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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "In this notebook we use `bseries.jl` to investigate error expansions for RK methods applied to specific problems." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 1, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "# Load the packages we will use. These must first be installed using: import Pkg; Pkg.add(\"package_name\")\n", | |
| "using BSeries\n", | |
| "using Latexify\n", | |
| "using RootedTrees\n", | |
| "using Symbolics\n", | |
| "import SymPy; sp=SymPy;" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "First we specify the Butcher coefficients of the RK method. This can include symbolic expressions and parameterized families of methods." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{h^{2}}{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{h^{3}}{8 \\alpha} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{h^{2}}{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{h^{3}}{8 \\alpha} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{}$\"" | |
| ] | |
| }, | |
| "execution_count": 2, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "α = sp.symbols(\"α\", real=true)\n", | |
| "A = [0 0; 1/(2*α) 0]; b = [1-α, α]; c = [0, 1/(2*α)]\n", | |
| "coeffs = bseries(A,b,c,3)\n", | |
| "latexify(coeffs, cdot=false)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{1}{2} h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + \\frac{1}{8} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + \\frac{1}{16} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + \\frac{-21246894637}{49670350848} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + \\frac{-722476128287}{1390769823744} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + \\frac{1970748171909370823}{42730909364772720000} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + \\frac{3898363669}{40242182400} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{1}{2} h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + \\frac{1}{8} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + \\frac{1}{16} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + \\frac{-21246894637}{49670350848} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + \\frac{-722476128287}{1390769823744} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + \\frac{1970748171909370823}{42730909364772720000} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + \\frac{3898363669}{40242182400} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$\"" | |
| ] | |
| }, | |
| "execution_count": 3, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "A = Rational{Int128}[0 0 0 0 0 0 0 0;(-1//6) (1//2) 0 0 0 0 0 0;(-1//10) (1//10) (1//2) 0 0 0 0 0;(-21463//39375) (21017//26250) (-5//9) (1//2) 0 0 0 0;(-59588//54675) (118717//36450) (-4375//2187) 0 (1//2) 0 0 0;(-19993033//9443328) (28508695//3147776) (-13577105//2360832) (-4090625//3147776) (1136025//3147776) (1//2) 0 0;(367020141781//199294617600) (814214904871//22143846400) (-29834937659//1992946176) (-1983358776875//87689631744) (-6702625935//885753856) (688576//109395) (1//2) 0;(1081252805//134140608) (2639189439//74522560) (33646441//4191894) (-7873511875//210792384) (-504040617//14904512) (2110843561//115277085) (13//7) (1//2)];\n", | |
| "b = Rational{Int128}[(1081252805//134140608),(2639189439//74522560),(33646441//4191894),(-7873511875//210792384),(-504040617//14904512),(2110843561//115277085),(13//7),(1//2)];\n", | |
| "c = Rational{Int128}[0,(1//3),(1//2),(1//5),(2//3),(3//4),(1//4),1];\n", | |
| "\n", | |
| "coeffs = bseries(A,b,c,5)\n", | |
| "latexify(coeffs, cdot=false)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Since $f$ has not been specified, the elementary differentials are indicated by the corresponding rooted tree. The rooted trees are printed as nested lists, essentially in the form used in Butcher's book. We can also print out the B-series coefficients this way:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "TruncatedBSeries{RootedTree{Int64, Vector{Int64}}, Rational{Int128}} with 18 entries:\n", | |
| " RootedTree{Int64}: Int64[] => 1//1\n", | |
| " RootedTree{Int64}: [1] => 1//1\n", | |
| " RootedTree{Int64}: [1, 2] => 1//2\n", | |
| " RootedTree{Int64}: [1, 2, 3] => 1//6\n", | |
| " RootedTree{Int64}: [1, 2, 2] => 1//3\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4] => 1//24\n", | |
| " RootedTree{Int64}: [1, 2, 3, 3] => 1//12\n", | |
| " RootedTree{Int64}: [1, 2, 3, 2] => 1//8\n", | |
| " RootedTree{Int64}: [1, 2, 2, 2] => 1//4\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 5] => 1//48\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 4] => 1//24\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 3] => 1//16\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 2] => -21246894637//49670350848\n", | |
| " RootedTree{Int64}: [1, 2, 3, 3, 3] => 1//8\n", | |
| " RootedTree{Int64}: [1, 2, 3, 3, 2] => -722476128287//695384911872\n", | |
| " RootedTree{Int64}: [1, 2, 3, 2, 3] => 1970748171909370823//213654546823863600…\n", | |
| " RootedTree{Int64}: [1, 2, 3, 2, 2] => 3898363669//20121091200\n", | |
| " RootedTree{Int64}: [1, 2, 2, 2, 2] => 1//2" | |
| ] | |
| }, | |
| "execution_count": 4, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "coeffs" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "In this form, the rooted trees are printed as level sets. The corresponding coefficients are on the right. We can also get the B-series of the exact solution:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 5, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "TruncatedBSeries{RootedTree{Int64, Vector{Int64}}, Rational{Int128}} with 18 entries:\n", | |
| " RootedTree{Int64}: Int64[] => 1//1\n", | |
| " RootedTree{Int64}: [1] => 1//1\n", | |
| " RootedTree{Int64}: [1, 2] => 1//2\n", | |
| " RootedTree{Int64}: [1, 2, 3] => 1//6\n", | |
| " RootedTree{Int64}: [1, 2, 2] => 1//3\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4] => 1//24\n", | |
| " RootedTree{Int64}: [1, 2, 3, 3] => 1//12\n", | |
| " RootedTree{Int64}: [1, 2, 3, 2] => 1//8\n", | |
| " RootedTree{Int64}: [1, 2, 2, 2] => 1//4\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 5] => 1//120\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 4] => 1//60\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 3] => 1//40\n", | |
| " RootedTree{Int64}: [1, 2, 3, 4, 2] => 1//30\n", | |
| " RootedTree{Int64}: [1, 2, 3, 3, 3] => 1//20\n", | |
| " RootedTree{Int64}: [1, 2, 3, 3, 2] => 1//15\n", | |
| " RootedTree{Int64}: [1, 2, 3, 2, 3] => 1//20\n", | |
| " RootedTree{Int64}: [1, 2, 3, 2, 2] => 1//10\n", | |
| " RootedTree{Int64}: [1, 2, 2, 2, 2] => 1//5" | |
| ] | |
| }, | |
| "execution_count": 5, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "coeffs_ex = ExactSolution(coeffs)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 50, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + 0.5 h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + 0.125 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + 0.05 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$" | |
| ], | |
| "text/plain": [ | |
| "L\"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + 0.5 h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + 0.125 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + 0.05 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$\"" | |
| ] | |
| }, | |
| "execution_count": 50, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "latexify(coeffs_ex,cdot=false)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Next we define our ODE. For a non-autonomous ODE, it's most convenient to just add $t$ as an additional variable. That makes the code below look a bit funny." | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 8, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "f (generic function with 1 method)" | |
| ] | |
| }, | |
| "execution_count": 8, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "λ = SymPy.symbols(\"λ\", real=true)\n", | |
| "function f(du, u, params, t)\n", | |
| " uu, tt = u\n", | |
| " du[1] = λ*(uu-sin(tt)) + sqrt(1-uu^2); du[2] = 1\n", | |
| " return nothing\n", | |
| "end" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Then we define a symbolic RHS:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 25, | |
| "metadata": {}, | |
| "outputs": [], | |
| "source": [ | |
| "Δt = sp.symbols(\"Δt\", real=true)\n", | |
| "u, t = sp.symbols(\"u t\", real=true)\n", | |
| "u_sym = [u, t]\n", | |
| "f_sym = similar(u_sym); f(f_sym, u_sym, nothing, nothing)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Finally, we get the B-Series for our RK method applied to our ODE:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 26, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\left[ \\begin{array}{r}\\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{3} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{48} + \\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{2} \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{48} - \\frac{18142497709 Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{49670350848} + \\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{48} - \\frac{722476128287 Δt^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{1390769823744} + \\frac{1970748171909370823 Δt^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2}}{42730909364772720000} + \\frac{3898363669 Δt^{5} \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{40242182400} + \\frac{Δt^{5} \\left(- λ \\sin{\\left(t \\right)} + \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(- \\frac{15 u^{4}}{\\left(1 - u^{2}\\right)^{\\frac{7}{2}}} - \\frac{18 u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right)\\right)}{48} + \\frac{Δt^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{24} + \\frac{Δt^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{24} + \\frac{Δt^{4} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{8} + \\frac{Δt^{4} \\left(λ \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{24} + \\frac{Δt^{3} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{6} + \\frac{Δt^{3} \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{6} + \\frac{Δt^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{2} + Δt \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) + 1\\\\Δt + 1\\end{array} \\right]$\n" | |
| ], | |
| "text/plain": [ | |
| "2-element Vector{SymPy.Sym}:\n", | |
| " Δt^5*(-u/sqrt(1 - u^2) + λ)^3*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/48 + Δt^5*(-u/sqrt(1 - u^2) + λ)^2*(λ*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^2)/48 - 18142497709*Δt^5*(-u/sqrt(1 - u^2) + λ)*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/49670350848 + Δt^5*(-u/sqrt(1 - u^2) + λ)*(λ*cos(t) + (-3*u^3/(1 - u^2)^(5/2) - 3*u/(1 - u^2)^(3/2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^3)/48 - 722476128287*Δt^5*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))*(λ*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^2)/1390769823744 + 1970748171909370823*Δt^5*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))^2/42730909364772720000 + 3898363669*Δt^5*(-3*u^3/(1 - u^2)^(5/2) - 3*u/(1 - u^2)^(3/2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^2*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/40242182400 + Δt^5*(-λ*sin(t) + (λ*(u - sin(t)) + sqrt(1 - u^2))^4*(-15*u^4/(1 - u^2)^(7/2) - 18*u^2/(1 - u^2)^(5/2) - 3/(1 - u^2)^(3/2)))/48 + Δt^4*(-u/sqrt(1 - u^2) + λ)^2*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/24 + Δt^4*(-u/sqrt(1 - u^2) + λ)*(λ*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^2)/24 + Δt^4*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/8 + Δt^4*(λ*cos(t) + (-3*u^3/(1 - u^2)^(5/2) - 3*u/(1 - u^2)^(3/2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^3)/24 + Δt^3*(-u/sqrt(1 - u^2) + λ)*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/6 + Δt^3*(λ*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(λ*(u - sin(t)) + sqrt(1 - u^2))^2)/6 + Δt^2*(-λ*cos(t) + (-u/sqrt(1 - u^2) + λ)*(λ*(u - sin(t)) + sqrt(1 - u^2)))/2 + Δt*(λ*(u - sin(t)) + sqrt(1 - u^2)) + 1\n", | |
| " Δt + 1" | |
| ] | |
| }, | |
| "execution_count": 26, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "evaluate(f_sym,u_sym,Δt,coeffs)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Here's the B-Series for the exact solution of the same ODE:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 27, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\begin{equation*}\\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{3} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{120} + \\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{2} \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{120} + \\frac{7 Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{120} + \\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{120} + \\frac{Δt^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{30} + \\frac{Δt^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2}}{40} + \\frac{Δt^{5} \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{20} + \\frac{Δt^{5} \\left(- λ \\sin{\\left(t \\right)} + \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(- \\frac{15 u^{4}}{\\left(1 - u^{2}\\right)^{\\frac{7}{2}}} - \\frac{18 u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right)\\right)}{120} + \\frac{Δt^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{24} + \\frac{Δt^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{24} + \\frac{Δt^{4} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{8} + \\frac{Δt^{4} \\left(λ \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{24} + \\frac{Δt^{3} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{6} + \\frac{Δt^{3} \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{6} + \\frac{Δt^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{2} + Δt \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) + 1\\end{equation*}$\n" | |
| ], | |
| "text/plain": [ | |
| " \n", | |
| " 3 ⎛ ⎛ \n", | |
| " 5 ⎛ u ⎞ ⎜ ⎛ u ⎞ ⎜ \n", | |
| "Δt ⋅⎜- ─────────── + λ⎟ ⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱\n", | |
| " ⎜ ________ ⎟ ⎜ ⎜ ________ ⎟ \n", | |
| " ⎜ ╱ 2 ⎟ ⎜ ⎜ ╱ 2 ⎟ \n", | |
| " ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 120 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ________⎞⎞ 2 ⎜ ⎛ 2 \n", | |
| "╱ 2 ⎟⎟ 5 ⎛ u ⎞ ⎜ ⎜ u 1 \n", | |
| " 1 - u ⎠⎟ Δt ⋅⎜- ─────────── + λ⎟ ⋅⎜λ⋅sin(t) + ⎜- ─────────── - ──────────\n", | |
| " ⎟ ⎜ ________ ⎟ ⎜ ⎜ 3/2 _______\n", | |
| " ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎜ ⎛ 2⎞ ╱ 2\n", | |
| " ⎠ ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ ╲╱ 1 - u \n", | |
| "─────────── + ────────────────────────────────────────────────────────────────\n", | |
| " 120 \n", | |
| "\n", | |
| " 2⎞ \n", | |
| " ⎞ ⎛ ________⎞ ⎟ ⎛ 2 \n", | |
| " ⎟ ⎜ ╱ 2 ⎟ ⎟ 5 ⎛ u ⎞ ⎜ u \n", | |
| "─⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ 7⋅Δt ⋅⎜- ─────────── + λ⎟⋅⎜- ───────────\n", | |
| "_⎟ ⎟ ⎜ ________ ⎟ ⎜ 3/2\n", | |
| " ⎟ ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎛ 2⎞ \n", | |
| " ⎠ ⎠ ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝1 - u ⎠ \n", | |
| "─────────────────────────────────── + ────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎞ ⎛ ________⎞ ⎛ \n", | |
| " 1 ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎛ u \n", | |
| " - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ\n", | |
| " ________⎟ ⎜ ⎜ ________ \n", | |
| " ╱ 2 ⎟ ⎜ ⎜ ╱ 2 \n", | |
| " ╲╱ 1 - u ⎠ ⎝ ⎝ ╲╱ 1 - u \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 120 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎛ ________⎞⎞ ⎜ ⎛ \n", | |
| "⎞ ⎜ ╱ 2 ⎟⎟ 5 ⎛ u ⎞ ⎜ ⎜ \n", | |
| "⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ Δt ⋅⎜- ─────────── + λ⎟⋅⎜λ⋅cos(t) + ⎜- ───\n", | |
| "⎟ ⎟ ⎜ ________ ⎟ ⎜ ⎜ \n", | |
| "⎟ ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎜ ⎛ \n", | |
| "⎠ ⎠ ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ⎝1 \n", | |
| "───────────────────────────────── + ──────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " 3⎞ \n", | |
| " 3 ⎞ ⎛ ________⎞ ⎟ ⎛ 2 \n", | |
| " 3⋅u 3⋅u ⎟ ⎜ ╱ 2 ⎟ ⎟ 5 ⎜ u \n", | |
| "──────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ Δt ⋅⎜- ─────────── \n", | |
| " 5/2 3/2⎟ ⎟ ⎜ 3/2 \n", | |
| " 2⎞ ⎛ 2⎞ ⎟ ⎟ ⎜ ⎛ 2⎞ \n", | |
| "- u ⎠ ⎝1 - u ⎠ ⎠ ⎠ ⎝ ⎝1 - u ⎠ \n", | |
| "──────────────────────────────────────────────────────── + ───────────────────\n", | |
| " 120 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎞ ⎛ ________⎞ ⎜ ⎛ 2 \n", | |
| " 1 ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎜ u \n", | |
| "- ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⋅⎜λ⋅sin(t) + ⎜- ─────────── - ───\n", | |
| " ________⎟ ⎜ ⎜ 3/2 \n", | |
| " ╱ 2 ⎟ ⎜ ⎜ ⎛ 2⎞ ╱\n", | |
| " ╲╱ 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ ╲╱ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 30 \n", | |
| "\n", | |
| " 2⎞ \n", | |
| " ⎞ ⎛ ________⎞ ⎟ ⎛ 2 ⎞\n", | |
| " 1 ⎟ ⎜ ╱ 2 ⎟ ⎟ 5 ⎜ u 1 ⎟\n", | |
| "────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ Δt ⋅⎜- ─────────── - ───────────⎟\n", | |
| "________⎟ ⎟ ⎜ 3/2 ________⎟\n", | |
| " 2 ⎟ ⎟ ⎜ ⎛ 2⎞ ╱ 2 ⎟\n", | |
| " 1 - u ⎠ ⎠ ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠\n", | |
| "────────────────────────────────────────── + ─────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " 2 \n", | |
| " ⎛ ⎛ ________⎞⎞ ⎛ \n", | |
| " ⎜ ⎛ u ⎞ ⎜ ╱ 2 ⎟⎟ 5 ⎜ \n", | |
| "⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ Δt ⋅⎜- ──\n", | |
| " ⎜ ⎜ ________ ⎟ ⎟ ⎜ \n", | |
| " ⎜ ⎜ ╱ 2 ⎟ ⎟ ⎜ ⎛ \n", | |
| " ⎝ ⎝ ╲╱ 1 - u ⎠ ⎠ ⎝ ⎝1\n", | |
| "────────────────────────────────────────────────────────────────── + ─────────\n", | |
| " 40 \n", | |
| "\n", | |
| " 2 \n", | |
| " 3 ⎞ ⎛ ________⎞ ⎛ \n", | |
| " 3⋅u 3⋅u ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎛ \n", | |
| "───────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⋅⎜-λ⋅cos(t) + ⎜- ─────\n", | |
| " 5/2 3/2⎟ ⎜ ⎜ __\n", | |
| " 2⎞ ⎛ 2⎞ ⎟ ⎜ ⎜ ╱ \n", | |
| " - u ⎠ ⎝1 - u ⎠ ⎠ ⎝ ⎝ ╲╱ 1\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 20 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎛ ________⎞⎞ ⎜ ⎛ \n", | |
| "u ⎞ ⎜ ╱ 2 ⎟⎟ 5 ⎜ ⎜ \n", | |
| "────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ Δt ⋅⎜-λ⋅sin(t) + ⎝λ⋅(u - sin(t))\n", | |
| "______ ⎟ ⎟ ⎜ \n", | |
| " 2 ⎟ ⎟ ⎜ \n", | |
| " - u ⎠ ⎠ ⎝ \n", | |
| "─────────────────────────────────────────── + ────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " 4 ⎞ \n", | |
| " ________⎞ ⎛ 4 2 ⎞⎟ \n", | |
| " ╱ 2 ⎟ ⎜ 15⋅u 18⋅u 3 ⎟⎟ 4 ⎛ u \n", | |
| " + ╲╱ 1 - u ⎠ ⋅⎜- ─────────── - ─────────── - ───────────⎟⎟ Δt ⋅⎜- ───────\n", | |
| " ⎜ 7/2 5/2 3/2⎟⎟ ⎜ ____\n", | |
| " ⎜ ⎛ 2⎞ ⎛ 2⎞ ⎛ 2⎞ ⎟⎟ ⎜ ╱ \n", | |
| " ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠⎠ ⎝ ╲╱ 1 -\n", | |
| "───────────────────────────────────────────────────────────── + ──────────────\n", | |
| " 120 \n", | |
| "\n", | |
| " \n", | |
| " 2 ⎛ ⎛ ________⎞⎞ \n", | |
| " ⎞ ⎜ ⎛ u ⎞ ⎜ ╱ 2 ⎟⎟ \n", | |
| "──── + λ⎟ ⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ \n", | |
| "____ ⎟ ⎜ ⎜ ________ ⎟ ⎟ \n", | |
| " 2 ⎟ ⎜ ⎜ ╱ 2 ⎟ ⎟ \n", | |
| " u ⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ ⎠ \n", | |
| "─────────────────────────────────────────────────────────────────────────── + \n", | |
| " 24 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎜ ⎛ 2 ⎞ ⎛ \n", | |
| " 4 ⎛ u ⎞ ⎜ ⎜ u 1 ⎟ ⎜ \n", | |
| "Δt ⋅⎜- ─────────── + λ⎟⋅⎜λ⋅sin(t) + ⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(\n", | |
| " ⎜ ________ ⎟ ⎜ ⎜ 3/2 ________⎟ \n", | |
| " ⎜ ╱ 2 ⎟ ⎜ ⎜ ⎛ 2⎞ ╱ 2 ⎟ \n", | |
| " ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 24 \n", | |
| "\n", | |
| " 2⎞ \n", | |
| " ________⎞ ⎟ ⎛ 2 ⎞ ⎛ \n", | |
| " ╱ 2 ⎟ ⎟ 4 ⎜ u 1 ⎟ ⎜ ╱\n", | |
| "t)) + ╲╱ 1 - u ⎠ ⎟ Δt ⋅⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ \n", | |
| " ⎟ ⎜ 3/2 ________⎟ \n", | |
| " ⎟ ⎜ ⎛ 2⎞ ╱ 2 ⎟ \n", | |
| " ⎠ ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠ \n", | |
| "──────────────────── + ───────────────────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| "________⎞ ⎛ ⎛ ________⎞⎞ \n", | |
| " 2 ⎟ ⎜ ⎛ u ⎞ ⎜ ╱ 2 ⎟⎟ \n", | |
| " 1 - u ⎠⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ Δ\n", | |
| " ⎜ ⎜ ________ ⎟ ⎟ \n", | |
| " ⎜ ⎜ ╱ 2 ⎟ ⎟ \n", | |
| " ⎝ ⎝ ╲╱ 1 - u ⎠ ⎠ \n", | |
| "────────────────────────────────────────────────────────────────────────── + ─\n", | |
| " 8 \n", | |
| "\n", | |
| " ⎛ 3⎞ \n", | |
| " ⎜ ⎛ 3 ⎞ ⎛ ________⎞ ⎟ \n", | |
| " 4 ⎜ ⎜ 3⋅u 3⋅u ⎟ ⎜ ╱ 2 ⎟ ⎟ \n", | |
| "t ⋅⎜λ⋅cos(t) + ⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ \n", | |
| " ⎜ ⎜ 5/2 3/2⎟ ⎟ \n", | |
| " ⎜ ⎜ ⎛ 2⎞ ⎛ 2⎞ ⎟ ⎟ \n", | |
| " ⎝ ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠ ⎠ \n", | |
| "───────────────────────────────────────────────────────────────────────────── \n", | |
| " 24 \n", | |
| "\n", | |
| " \n", | |
| " ⎛ ⎛ \n", | |
| " 3 ⎛ u ⎞ ⎜ ⎛ u ⎞ ⎜ \n", | |
| " Δt ⋅⎜- ─────────── + λ⎟⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲\n", | |
| " ⎜ ________ ⎟ ⎜ ⎜ ________ ⎟ \n", | |
| " ⎜ ╱ 2 ⎟ ⎜ ⎜ ╱ 2 ⎟ \n", | |
| " ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ \n", | |
| "+ ────────────────────────────────────────────────────────────────────────────\n", | |
| " 6 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ________⎞⎞ ⎜ ⎛ 2 ⎞ ⎛ \n", | |
| " ╱ 2 ⎟⎟ 3 ⎜ ⎜ u 1 ⎟ ⎜ \n", | |
| "╱ 1 - u ⎠⎟ Δt ⋅⎜λ⋅sin(t) + ⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) +\n", | |
| " ⎟ ⎜ ⎜ 3/2 ________⎟ \n", | |
| " ⎟ ⎜ ⎜ ⎛ 2⎞ ╱ 2 ⎟ \n", | |
| " ⎠ ⎝ ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠ \n", | |
| "──────────── + ───────────────────────────────────────────────────────────────\n", | |
| " 6 \n", | |
| "\n", | |
| " 2⎞ \n", | |
| " ________⎞ ⎟ ⎛ ⎛ __\n", | |
| " ╱ 2 ⎟ ⎟ 2 ⎜ ⎛ u ⎞ ⎜ ╱ \n", | |
| " ╲╱ 1 - u ⎠ ⎟ Δt ⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1\n", | |
| " ⎟ ⎜ ⎜ ________ ⎟ \n", | |
| " ⎟ ⎜ ⎜ ╱ 2 ⎟ \n", | |
| " ⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ \n", | |
| "─────────────── + ────────────────────────────────────────────────────────────\n", | |
| " 2 \n", | |
| "\n", | |
| " \n", | |
| "______⎞⎞ \n", | |
| " 2 ⎟⎟ \n", | |
| " - u ⎠⎟ \n", | |
| " ⎟ \n", | |
| " ⎟ ⎛ ________⎞ \n", | |
| " ⎠ ⎜ ╱ 2 ⎟ \n", | |
| "──────── + Δt⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ + 1\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 27, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "evaluate(f_sym,u_sym,Δt,coeffs_ex)[1]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "And their difference, which is the local error:" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 28, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\begin{equation*}\\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{3} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{80} + \\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right)^{2} \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{80} - \\frac{105199674209 Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{248351754240} + \\frac{Δt^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{80} - \\frac{3844175612059 Δt^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(λ \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{6953849118720} + \\frac{902475437790052823 Δt^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2}}{42730909364772720000} + \\frac{1886254549 Δt^{5} \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(- λ \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + λ\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{40242182400} + \\frac{Δt^{5} \\left(- λ \\sin{\\left(t \\right)} + \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(- \\frac{15 u^{4}}{\\left(1 - u^{2}\\right)^{\\frac{7}{2}}} - \\frac{18 u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right)\\right)}{80}\\end{equation*}$\n" | |
| ], | |
| "text/plain": [ | |
| " \n", | |
| " 3 ⎛ ⎛ \n", | |
| " 5 ⎛ u ⎞ ⎜ ⎛ u ⎞ ⎜ \n", | |
| "Δt ⋅⎜- ─────────── + λ⎟ ⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱\n", | |
| " ⎜ ________ ⎟ ⎜ ⎜ ________ ⎟ \n", | |
| " ⎜ ╱ 2 ⎟ ⎜ ⎜ ╱ 2 ⎟ \n", | |
| " ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 80 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ________⎞⎞ 2 ⎜ ⎛ 2 \n", | |
| "╱ 2 ⎟⎟ 5 ⎛ u ⎞ ⎜ ⎜ u 1 \n", | |
| " 1 - u ⎠⎟ Δt ⋅⎜- ─────────── + λ⎟ ⋅⎜λ⋅sin(t) + ⎜- ─────────── - ──────────\n", | |
| " ⎟ ⎜ ________ ⎟ ⎜ ⎜ 3/2 _______\n", | |
| " ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎜ ⎛ 2⎞ ╱ 2\n", | |
| " ⎠ ⎝ ╲╱ 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ ╲╱ 1 - u \n", | |
| "─────────── + ────────────────────────────────────────────────────────────────\n", | |
| " 80 \n", | |
| "\n", | |
| " 2⎞ \n", | |
| " ⎞ ⎛ ________⎞ ⎟ ⎛ \n", | |
| " ⎟ ⎜ ╱ 2 ⎟ ⎟ 5 ⎛ u ⎞ ⎜ \n", | |
| "─⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ 105199674209⋅Δt ⋅⎜- ─────────── + λ⎟⋅⎜- \n", | |
| "_⎟ ⎟ ⎜ ________ ⎟ ⎜ \n", | |
| " ⎟ ⎟ ⎜ ╱ 2 ⎟ ⎜ \n", | |
| " ⎠ ⎠ ⎝ ╲╱ 1 - u ⎠ ⎝ \n", | |
| "─────────────────────────────────── - ────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 2 ⎞ ⎛ ________⎞ ⎛ \n", | |
| " u 1 ⎟ ⎜ ╱ 2 ⎟ ⎜ ⎛ \n", | |
| "─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⋅⎜-λ⋅cos(t) + ⎜- ────\n", | |
| " 3/2 ________⎟ ⎜ ⎜ _\n", | |
| "⎛ 2⎞ ╱ 2 ⎟ ⎜ ⎜ ╱ \n", | |
| "⎝1 - u ⎠ ╲╱ 1 - u ⎠ ⎝ ⎝ ╲╱ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 248351754240 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎛ ________⎞⎞ ⎜ \n", | |
| " u ⎞ ⎜ ╱ 2 ⎟⎟ 5 ⎛ u ⎞ ⎜ \n", | |
| "─────── + λ⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ Δt ⋅⎜- ─────────── + λ⎟⋅⎜λ⋅cos(\n", | |
| "_______ ⎟ ⎟ ⎜ ________ ⎟ ⎜ \n", | |
| " 2 ⎟ ⎟ ⎜ ╱ 2 ⎟ ⎜ \n", | |
| "1 - u ⎠ ⎠ ⎝ ╲╱ 1 - u ⎠ ⎝ \n", | |
| "──────────────────────────────────────────── + ───────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " 3⎞ \n", | |
| " ⎛ 3 ⎞ ⎛ ________⎞ ⎟ \n", | |
| " ⎜ 3⋅u 3⋅u ⎟ ⎜ ╱ 2 ⎟ ⎟ \n", | |
| "t) + ⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ 38441756\n", | |
| " ⎜ 5/2 3/2⎟ ⎟ \n", | |
| " ⎜ ⎛ 2⎞ ⎛ 2⎞ ⎟ ⎟ \n", | |
| " ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠ ⎠ \n", | |
| "─────────────────────────────────────────────────────────────────── - ────────\n", | |
| " 80 \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎛ 2 ⎞ ⎛ ________⎞ ⎜ \n", | |
| " 5 ⎜ u 1 ⎟ ⎜ ╱ 2 ⎟ ⎜ \n", | |
| "12059⋅Δt ⋅⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⋅⎜λ⋅sin(\n", | |
| " ⎜ 3/2 ________⎟ ⎜ \n", | |
| " ⎜ ⎛ 2⎞ ╱ 2 ⎟ ⎜ \n", | |
| " ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠ ⎝ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 6953849118720 \n", | |
| "\n", | |
| " 2⎞ \n", | |
| " ⎛ 2 ⎞ ⎛ ________⎞ ⎟ \n", | |
| " ⎜ u 1 ⎟ ⎜ ╱ 2 ⎟ ⎟ \n", | |
| "t) + ⎜- ─────────── - ───────────⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎟ 90247543\n", | |
| " ⎜ 3/2 ________⎟ ⎟ \n", | |
| " ⎜ ⎛ 2⎞ ╱ 2 ⎟ ⎟ \n", | |
| " ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠ ⎠ \n", | |
| "─────────────────────────────────────────────────────────────────── + ────────\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎛ 2 ⎞ ⎛ \n", | |
| " 5 ⎜ u 1 ⎟ ⎜ ⎛ u ⎞ \n", | |
| "7790052823⋅Δt ⋅⎜- ─────────── - ───────────⎟⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅\n", | |
| " ⎜ 3/2 ________⎟ ⎜ ⎜ ________ ⎟ \n", | |
| " ⎜ ⎛ 2⎞ ╱ 2 ⎟ ⎜ ⎜ ╱ 2 ⎟ \n", | |
| " ⎝ ⎝1 - u ⎠ ╲╱ 1 - u ⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 42730909364772720000 \n", | |
| "\n", | |
| " 2 \n", | |
| "⎛ ________⎞⎞ ⎛ 3 \n", | |
| "⎜ ╱ 2 ⎟⎟ 5 ⎜ 3⋅u 3⋅u \n", | |
| "⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎟ 1886254549⋅Δt ⋅⎜- ─────────── - ───────────\n", | |
| " ⎟ ⎜ 5/2 3/2\n", | |
| " ⎟ ⎜ ⎛ 2⎞ ⎛ 2⎞ \n", | |
| " ⎠ ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ \n", | |
| "──────────────────────────────── + ───────────────────────────────────────────\n", | |
| " \n", | |
| "\n", | |
| " 2 \n", | |
| "⎞ ⎛ ________⎞ ⎛ ⎛ \n", | |
| "⎟ ⎜ ╱ 2 ⎟ ⎜ ⎛ u ⎞ ⎜ \n", | |
| "⎟⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⋅⎜-λ⋅cos(t) + ⎜- ─────────── + λ⎟⋅⎝λ⋅(u - sin\n", | |
| "⎟ ⎜ ⎜ ________ ⎟ \n", | |
| "⎟ ⎜ ⎜ ╱ 2 ⎟ \n", | |
| "⎠ ⎝ ⎝ ╲╱ 1 - u ⎠ \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 40242182400 \n", | |
| "\n", | |
| " ⎛ 4 \n", | |
| " ________⎞⎞ ⎜ ⎛ ________⎞ ⎛ \n", | |
| " ╱ 2 ⎟⎟ 5 ⎜ ⎜ ╱ 2 ⎟ ⎜ \n", | |
| "(t)) + ╲╱ 1 - u ⎠⎟ Δt ⋅⎜-λ⋅sin(t) + ⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⋅⎜- ───\n", | |
| " ⎟ ⎜ ⎜ \n", | |
| " ⎟ ⎜ ⎜ ⎛ \n", | |
| " ⎠ ⎝ ⎝ ⎝1 \n", | |
| "──────────────────── + ───────────────────────────────────────────────────────\n", | |
| " 80 \n", | |
| "\n", | |
| " ⎞\n", | |
| " 4 2 ⎞⎟\n", | |
| "15⋅u 18⋅u 3 ⎟⎟\n", | |
| "──────── - ─────────── - ───────────⎟⎟\n", | |
| " 7/2 5/2 3/2⎟⎟\n", | |
| " 2⎞ ⎛ 2⎞ ⎛ 2⎞ ⎟⎟\n", | |
| "- u ⎠ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠⎠\n", | |
| "──────────────────────────────────────\n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 28, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "expr = simplify(evaluate(f_sym,u_sym,Δt,coeffs)-evaluate(f_sym,u_sym,Δt,coeffs_ex))[1]" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 36, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\begin{equation*}\\frac{4535459777238976256125 \\left(1 - u^{2}\\right)^{\\frac{35}{2}} \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(λ \\left(1 - u^{2}\\right)^{2} \\sin{\\left(t \\right)} - \\sqrt{1 - u^{2}} \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right) + 611621992800 \\left(1 - u^{2}\\right)^{\\frac{33}{2}} \\left(1886254549 u \\left(1 - u^{2}\\right)^{\\frac{3}{2}} \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(λ \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - λ \\sqrt{1 - u^{2}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right) + 167675760 \\left(- u + λ \\sqrt{1 - u^{2}}\\right) \\left(- 3 u \\left(1 - u^{2}\\right)^{\\frac{3}{2}} \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3} + λ \\left(1 - u^{2}\\right)^{4} \\cos{\\left(t \\right)}\\right)\\right) - 102554182475454528000 \\left(1 - u^{2}\\right)^{\\frac{27}{2}} \\left(λ \\left(1 - u^{2}\\right)^{\\frac{15}{2}} \\sin{\\left(t \\right)} + 3 \\left(1 - u^{2}\\right)^{4} \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(5 u^{4} + 6 u^{2} \\left(1 - u^{2}\\right) + \\left(1 - u^{2}\\right)^{2}\\right)\\right) + 102554182475454528000 \\left(1 - u^{2}\\right)^{19} \\left(u - λ \\sqrt{1 - u^{2}}\\right)^{3} \\left(λ \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - λ \\sqrt{1 - u^{2}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right) + 4 \\left(1 - u^{2}\\right)^{18} \\left(868821451912298236625 \\sqrt{1 - u^{2}} \\left(u - λ \\sqrt{1 - u^{2}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(λ \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - λ \\sqrt{1 - u^{2}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right) - 43318821013922535504 \\sqrt{1 - u^{2}} \\left(λ \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - λ \\sqrt{1 - u^{2}}\\right) \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2} + 25638545618863632000 \\left(u - λ \\sqrt{1 - u^{2}}\\right)^{2} \\left(λ \\left(1 - u^{2}\\right)^{2} \\sin{\\left(t \\right)} - \\sqrt{1 - u^{2}} \\left(λ \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)\\right)}{8204334598036362240000 \\left(1 - u^{2}\\right)^{21}}\\end{equation*}$\n" | |
| ], | |
| "text/plain": [ | |
| " ⎛ \n", | |
| " 35/2 ⎛ ________⎞ ⎜ \n", | |
| " ⎛ 2⎞ ⎜ ╱ 2 ⎟ ⎜ ⎛ 2⎞\n", | |
| "4535459777238976256125⋅⎝1 - u ⎠ ⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⋅⎝λ⋅⎝1 - u ⎠\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 2⎞ \n", | |
| "2 ________ ⎛ ________⎞ ⎟ \n", | |
| " ╱ 2 ⎜ ╱ 2 ⎟ ⎟ ⎛ 2\n", | |
| " ⋅sin(t) - ╲╱ 1 - u ⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎠ + 611621992800⋅⎝1 - u \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " ⎛ 2 \n", | |
| " 33/2 ⎜ 3/2 ⎛ ________⎞ ⎛ ________\n", | |
| "⎞ ⎜ ⎛ 2⎞ ⎜ ╱ 2 ⎟ ⎜ ╱ 2 \n", | |
| "⎠ ⋅⎝1886254549⋅u⋅⎝1 - u ⎠ ⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⋅⎝λ⋅╲╱ 1 - u \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ⎛ ________⎞ ⎛ ________⎞⎞ ⎛ \n", | |
| " ⎜ ╱ 2 ⎟ ⎜ ╱ 2 ⎟⎟ ⎜ \n", | |
| "⋅cos(t) + ⎝u - λ⋅╲╱ 1 - u ⎠⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎠ + 167675760⋅⎝-u \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " ⎛ 3 \n", | |
| " ________⎞ ⎜ 3/2 ⎛ ________⎞ \n", | |
| " ╱ 2 ⎟ ⎜ ⎛ 2⎞ ⎜ ╱ 2 ⎟ ⎛ \n", | |
| "+ λ⋅╲╱ 1 - u ⎠⋅⎝- 3⋅u⋅⎝1 - u ⎠ ⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ + λ⋅⎝1 - u\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " ⎞⎞ ⎛ \n", | |
| " 4 ⎟⎟ 27/2 ⎜ 15/2 \n", | |
| "2⎞ ⎟⎟ ⎛ 2⎞ ⎜ ⎛ 2⎞ ⎛\n", | |
| " ⎠ ⋅cos(t)⎠⎠ - 102554182475454528000⋅⎝1 - u ⎠ ⋅⎝λ⋅⎝1 - u ⎠ ⋅sin(t) + 3⋅⎝\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 4 ⎞ \n", | |
| " 4 ⎛ ________⎞ ⎛ 2⎞⎟ \n", | |
| " 2⎞ ⎜ ╱ 2 ⎟ ⎜ 4 2 ⎛ 2⎞ ⎛ 2⎞ ⎟⎟ \n", | |
| "1 - u ⎠ ⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⋅⎝5⋅u + 6⋅u ⋅⎝1 - u ⎠ + ⎝1 - u ⎠ ⎠⎠ +\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " ⎛\n", | |
| " 8204334598036362240000⋅⎝\n", | |
| "\n", | |
| " 3 \n", | |
| " 19 ⎛ ________⎞ ⎛ ________ \n", | |
| " ⎛ 2⎞ ⎜ ╱ 2 ⎟ ⎜ ╱ 2 \n", | |
| " 102554182475454528000⋅⎝1 - u ⎠ ⋅⎝u - λ⋅╲╱ 1 - u ⎠ ⋅⎝λ⋅╲╱ 1 - u ⋅cos(t) +\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " 21 \n", | |
| " 2⎞ \n", | |
| "1 - u ⎠ \n", | |
| "\n", | |
| " ⎛ \n", | |
| " ⎛ ________⎞ ⎛ ________⎞⎞ 18 ⎜ \n", | |
| " ⎜ ╱ 2 ⎟ ⎜ ╱ 2 ⎟⎟ ⎛ 2⎞ ⎜ \n", | |
| " ⎝u - λ⋅╲╱ 1 - u ⎠⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎠ + 4⋅⎝1 - u ⎠ ⋅⎝868821451\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ________ ⎛ ________⎞ ⎛ ________⎞ ⎛ \n", | |
| " ╱ 2 ⎜ ╱ 2 ⎟ ⎜ ╱ 2 ⎟ ⎜ \n", | |
| "912298236625⋅╲╱ 1 - u ⋅⎝u - λ⋅╲╱ 1 - u ⎠⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⋅⎝λ\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ________ ⎛ ________⎞ ⎛ ________⎞⎞ \n", | |
| " ╱ 2 ⎜ ╱ 2 ⎟ ⎜ ╱ 2 ⎟⎟ \n", | |
| "⋅╲╱ 1 - u ⋅cos(t) + ⎝u - λ⋅╲╱ 1 - u ⎠⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠⎠ - 43\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " ________ ⎛ ________ ⎛ ________⎞ ⎛ \n", | |
| " ╱ 2 ⎜ ╱ 2 ⎜ ╱ 2 ⎟ ⎜ \n", | |
| "318821013922535504⋅╲╱ 1 - u ⋅⎝λ⋅╲╱ 1 - u ⋅cos(t) + ⎝u - λ⋅╲╱ 1 - u ⎠⋅⎝λ⋅\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 2 2 ⎛ \n", | |
| " ________⎞⎞ ⎛ ________⎞ ⎜ \n", | |
| " ╱ 2 ⎟⎟ ⎜ ╱ 2 ⎟ ⎜ ⎛\n", | |
| "(u - sin(t)) + ╲╱ 1 - u ⎠⎠ + 25638545618863632000⋅⎝u - λ⋅╲╱ 1 - u ⎠ ⋅⎝λ⋅⎝\n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " \n", | |
| "\n", | |
| " 2⎞⎞\n", | |
| " 2 ________ ⎛ ________⎞ ⎟⎟\n", | |
| " 2⎞ ╱ 2 ⎜ ╱ 2 ⎟ ⎟⎟\n", | |
| "1 - u ⎠ ⋅sin(t) - ╲╱ 1 - u ⋅⎝λ⋅(u - sin(t)) + ╲╱ 1 - u ⎠ ⎠⎠\n", | |
| "───────────────────────────────────────────────────────────────\n", | |
| " \n", | |
| " \n", | |
| " " | |
| ] | |
| }, | |
| "execution_count": 36, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "sp.simplify(expand(expr))" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 38, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\begin{equation*}- \\frac{3 u^{8} Δt^{5}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{6} Δt^{5} \\sqrt{1 - u^{2}}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3 u^{6} Δt^{5}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{6} Δt^{5}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} + \\frac{u^{6} Δt^{5}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} - \\frac{5407445509 u^{6} Δt^{5}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{4} Δt^{5} \\sqrt{1 - u^{2}}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3 u^{4} Δt^{5}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{4} Δt^{5}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} - \\frac{u^{4} Δt^{5}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} + \\frac{8425609189 u^{4} Δt^{5}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{328105563238046242399 u^{4} Δt^{5}}{241303958765775360000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{2} Δt^{5}}{40 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3635043306271280591533 u^{2} Δt^{5}}{4102167299018181120000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{754418498879 u^{2} Δt^{5}}{772649902080 \\sqrt{1 - u^{2}}} - \\frac{3 Δt^{5}}{80 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 Δt^{5}}{6953849118720 \\sqrt{1 - u^{2}}} + λ^{5} \\left(\\frac{u Δt^{5}}{80} - \\frac{Δt^{5} \\sin{\\left(t \\right)}}{80}\\right) + λ^{4} \\left(- \\frac{3 u^{8} Δt^{5}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3 u^{7} Δt^{5} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{6} Δt^{5} \\sin^{2}{\\left(t \\right)}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{5407445509 u^{6} Δt^{5}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3 u^{5} Δt^{5} \\sin^{3}{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{6413500069 u^{5} Δt^{5} \\sin{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3 u^{4} Δt^{5} \\sin^{4}{\\left(t \\right)}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{8425609189 u^{4} Δt^{5} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{357614120019468965321 u^{4} Δt^{5}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{14461936549 u^{3} Δt^{5} \\sin^{3}{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{98031318961151965421 u^{3} Δt^{5} \\sin{\\left(t \\right)}}{1025541824754545280000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{2} Δt^{5} \\sin^{4}{\\left(t \\right)}}{40 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{757632721070389930279 u^{2} Δt^{5} \\sin^{2}{\\left(t \\right)}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{697309902804057541121 u^{2} Δt^{5}}{2051083649509090560000 \\sqrt{1 - u^{2}}} + \\frac{4401390949 u Δt^{5} \\sin^{3}{\\left(t \\right)}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{748586994041784805121 u Δt^{5} \\sin{\\left(t \\right)}}{1025541824754545280000 \\sqrt{1 - u^{2}}} + \\frac{Δt^{5} \\sqrt{1 - u^{2}}}{80} - \\frac{Δt^{5} \\cos{\\left(t \\right)}}{80} - \\frac{3 Δt^{5} \\sin^{4}{\\left(t \\right)}}{80 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{799864085279512069121 Δt^{5} \\sin^{2}{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}}\\right) + λ^{3} \\left(- \\frac{3 u^{7} Δt^{5} \\sqrt{1 - u^{2}}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{7} Δt^{5}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3 u^{7} Δt^{5}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} + \\frac{1886254549 u^{7} Δt^{5}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} + \\frac{9 u^{6} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{6} Δt^{5} \\sin{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{9 u^{6} Δt^{5} \\sin{\\left(t \\right)}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} - \\frac{5658763647 u^{6} Δt^{5} \\sin{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{9 u^{5} Δt^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{6413500069 u^{5} Δt^{5} \\sqrt{1 - u^{2}}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{5} Δt^{5} \\sin^{2}{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{9 u^{5} Δt^{5} \\sin^{2}{\\left(t \\right)}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} + \\frac{5658763647 u^{5} Δt^{5} \\sin^{2}{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} + \\frac{1886254549 u^{5} Δt^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{5} Δt^{5}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{16341676594871 u^{5} Δt^{5}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{105199674209 u^{5} Δt^{5}}{124175877120 u^{4} - 248351754240 u^{2} + 124175877120} + \\frac{3 u^{5} Δt^{5}}{80 u^{4} - 160 u^{2} + 80} + \\frac{3 u^{4} Δt^{5} \\sqrt{1 - u^{2}} \\sin^{3}{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{8425609189 u^{4} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{2235676800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{4} Δt^{5} \\sin^{3}{\\left(t \\right)}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3 u^{4} Δt^{5} \\sin^{3}{\\left(t \\right)}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} - \\frac{1886254549 u^{4} Δt^{5} \\sin^{3}{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{1886254549 u^{4} Δt^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{10682727341193180000 u^{4} - 21365454682386360000 u^{2} + 10682727341193180000} + \\frac{49459645354533 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} + \\frac{105199674209 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{62087938560 u^{4} - 124175877120 u^{2} + 62087938560} - \\frac{9 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{80 u^{4} - 160 u^{2} + 80} - \\frac{14461936549 u^{3} Δt^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{98031318961151965421 u^{3} Δt^{5} \\sqrt{1 - u^{2}}}{1025541824754545280000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{1886254549 u^{3} Δt^{5} \\sin^{2}{\\left(t \\right)} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{3} Δt^{5} \\sin^{2}{\\left(t \\right)}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{49894260924453 u^{3} Δt^{5} \\sin^{2}{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{105199674209 u^{3} Δt^{5} \\sin^{2}{\\left(t \\right)}}{124175877120 u^{4} - 248351754240 u^{2} + 124175877120} + \\frac{9 u^{3} Δt^{5} \\sin^{2}{\\left(t \\right)}}{80 u^{4} - 160 u^{2} + 80} - \\frac{493765118337591853817 u^{3} Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{10319047364622345641093 u^{3} Δt^{5}}{8204334598036362240000 \\left(1 - u^{2}\\right)} + \\frac{9 u^{2} Δt^{5} \\sqrt{1 - u^{2}} \\sin^{3}{\\left(t \\right)}}{10 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{757632721070389930279 u^{2} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{1025541824754545280000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{16776292164791 u^{2} Δt^{5} \\sin^{3}{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{3 u^{2} Δt^{5} \\sin^{3}{\\left(t \\right)}}{80 u^{4} - 160 u^{2} + 80} + \\frac{12079201575925326001 u^{2} Δt^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{120651979382887680000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{25788879601336394706311 u^{2} Δt^{5} \\sin{\\left(t \\right)}}{8204334598036362240000 \\left(1 - u^{2}\\right)} - \\frac{4401390949 u Δt^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{748586994041784805121 u Δt^{5}}{1025541824754545280000} + \\frac{1886254549 u Δt^{5} \\sin^{2}{\\left(t \\right)} \\cos{\\left(t \\right)}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{20005292013953025321343 u Δt^{5} \\sin^{2}{\\left(t \\right)}}{8204334598036362240000 \\left(1 - u^{2}\\right)} - \\frac{705268173027862269617 u Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}} + \\frac{3 Δt^{5} \\sqrt{1 - u^{2}} \\sin^{3}{\\left(t \\right)}}{20 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{787044812470080253121 Δt^{5} \\sin{\\left(t \\right)}}{1025541824754545280000} + \\frac{3844175612059 Δt^{5} \\sin^{3}{\\left(t \\right)}}{6953849118720 \\left(1 - u^{2}\\right)} + \\frac{782183809884453165617 Δt^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}}\\right) + λ^{2} \\left(\\frac{9 u^{8} Δt^{5}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{7} Δt^{5} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{6} Δt^{5} \\sqrt{1 - u^{2}}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{9 u^{6} Δt^{5} \\sin^{2}{\\left(t \\right)}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{6} Δt^{5}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{6} Δt^{5}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} + \\frac{105199674209 u^{6} Δt^{5}}{248351754240 u^{4} \\sqrt{1 - u^{2}} - 496703508480 u^{2} \\sqrt{1 - u^{2}} + 248351754240 \\sqrt{1 - u^{2}}} - \\frac{u^{6} Δt^{5}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} + \\frac{5407445509 u^{6} Δt^{5}}{2235676800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{5} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{1158974853120 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{9 u^{5} Δt^{5} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{5} Δt^{5} \\sin{\\left(t \\right)}}{21365454682386360000 u^{4} \\sqrt{1 - u^{2}} - 42730909364772720000 u^{2} \\sqrt{1 - u^{2}} + 21365454682386360000 \\sqrt{1 - u^{2}}} - \\frac{105199674209 u^{5} Δt^{5} \\sin{\\left(t \\right)}}{124175877120 u^{4} \\sqrt{1 - u^{2}} - 248351754240 u^{2} \\sqrt{1 - u^{2}} + 124175877120 \\sqrt{1 - u^{2}}} + \\frac{2 u^{5} Δt^{5} \\sin{\\left(t \\right)}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} - \\frac{6413500069 u^{5} Δt^{5} \\sin{\\left(t \\right)}}{1490451200 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{4} Δt^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{1886254549 u^{4} Δt^{5} \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{4} Δt^{5} \\sin^{2}{\\left(t \\right)}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{4} Δt^{5} \\sin^{2}{\\left(t \\right)}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} + \\frac{105199674209 u^{4} Δt^{5} \\sin^{2}{\\left(t \\right)}}{248351754240 u^{4} \\sqrt{1 - u^{2}} - 496703508480 u^{2} \\sqrt{1 - u^{2}} + 248351754240 \\sqrt{1 - u^{2}}} - \\frac{u^{4} Δt^{5} \\sin^{2}{\\left(t \\right)}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} + \\frac{8425609189 u^{4} Δt^{5} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{8425609189 u^{4} Δt^{5}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{782183809884453165617 u^{4} Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 u^{4} - 4102167299018181120000 u^{2} + 2051083649509090560000} - \\frac{14903400795783417131743 u^{4} Δt^{5}}{4102167299018181120000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{1886254549 u^{3} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{14461936549 u^{3} Δt^{5} \\sin{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{782183809884453165617 u^{3} Δt^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{2051083649509090560000 u^{4} - 4102167299018181120000 u^{2} + 2051083649509090560000} + \\frac{48926120124603963505793 u^{3} Δt^{5} \\sin{\\left(t \\right)}}{8204334598036362240000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{12079201575925326001 u^{2} Δt^{5} \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)}}{120651979382887680000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{27 u^{2} Δt^{5} \\sin^{2}{\\left(t \\right)}}{20 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{22318774874155177518791 u^{2} Δt^{5} \\sin^{2}{\\left(t \\right)}}{8204334598036362240000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{2} Δt^{5} \\cos^{2}{\\left(t \\right)}}{42730909364772720000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{757632721070389930279 u^{2} Δt^{5}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{705268173027862269617 u^{2} Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(1 - u^{2}\\right)} - \\frac{5102610026961050007259 u^{2} Δt^{5}}{1640866919607272448000 \\sqrt{1 - u^{2}}} - \\frac{1886254549 u Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{6707030400 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{4401390949 u Δt^{5} \\sin{\\left(t \\right)}}{4471353600 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{782183809884453165617 u Δt^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(1 - u^{2}\\right)} + \\frac{14780311813398039280937 u Δt^{5} \\sin{\\left(t \\right)}}{2734778199345454080000 \\sqrt{1 - u^{2}}} + \\frac{105199674209 Δt^{5} \\sqrt{1 - u^{2}}}{248351754240} - \\frac{756545264265589533617 Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000} - \\frac{9 Δt^{5} \\sin^{2}{\\left(t \\right)}}{40 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 Δt^{5} \\sin^{2}{\\left(t \\right)}}{1738462279680 \\sqrt{1 - u^{2}}} - \\frac{902475437790052823 Δt^{5} \\cos^{2}{\\left(t \\right)}}{42730909364772720000 \\sqrt{1 - u^{2}}} - \\frac{1436611804849711823 Δt^{5}}{42730909364772720000 \\sqrt{1 - u^{2}}}\\right) + λ \\left(\\frac{3 u^{7} Δt^{5} \\sqrt{1 - u^{2}}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{7} Δt^{5}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3395336389 u^{7} Δt^{5}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{3 u^{6} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{6} Δt^{5} \\sin{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3395336389 u^{6} Δt^{5} \\sin{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{3 u^{5} Δt^{5} \\sqrt{1 - u^{2}}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{6078148549 u^{5} Δt^{5} \\sqrt{1 - u^{2}}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{5} Δt^{5}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3395336389 u^{5} Δt^{5}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{1886254549 u^{5} Δt^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{5} Δt^{5}}{10682727341193180000 u^{4} - 21365454682386360000 u^{2} + 10682727341193180000} + \\frac{3186582816619 u^{5} Δt^{5}}{1931624755200 u^{4} - 3863249510400 u^{2} + 1931624755200} + \\frac{u^{5} Δt^{5}}{80 \\left(u^{4} - 2 u^{2} + 1\\right)} + \\frac{3 u^{4} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{7922581909 u^{4} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3395336389 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} + \\frac{902475437790052823 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{3234873435499 u^{4} Δt^{5} \\sin{\\left(t \\right)}}{1931624755200 u^{4} - 3863249510400 u^{2} + 1931624755200} - \\frac{u^{4} Δt^{5} \\sin{\\left(t \\right)}}{80 \\left(u^{4} - 2 u^{2} + 1\\right)} - \\frac{14461936549 u^{3} Δt^{5} \\sqrt{1 - u^{2}}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{22917976411669 u^{3} Δt^{5} \\sqrt{1 - u^{2}}}{8692311398400 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{1886254549 u^{3} Δt^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{3} Δt^{5}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{14385906530231 u^{3} Δt^{5}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} + \\frac{519403663956455485817 u^{3} Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{9972496796510965357061 u^{3} Δt^{5}}{8204334598036362240000 \\left(1 - u^{2}\\right)} + \\frac{9 u^{2} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{10 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{86326046719459 u^{2} Δt^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{34769245593600 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{14820522100151 u^{2} Δt^{5} \\sin{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{10831818276999618281093 u^{2} Δt^{5} \\sin{\\left(t \\right)}}{8204334598036362240000 \\left(1 - u^{2}\\right)} - \\frac{1886254549 u Δt^{5} \\sqrt{1 - u^{2}}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{826854427199 u Δt^{5}}{386324951040} + \\frac{1886254549 u Δt^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{234341226128040406829 u Δt^{5}}{482607917531550720000 \\left(1 - u^{2}\\right)} + \\frac{756545264265589533617 u Δt^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}} + \\frac{4162893696667 Δt^{5} \\sin{\\left(t \\right)}}{2317949706240} + \\frac{3844175612059 Δt^{5} \\sin{\\left(t \\right)}}{6953849118720 \\left(1 - u^{2}\\right)}\\right)\\end{equation*}$\n" | |
| ], | |
| "text/plain": [ | |
| " \n", | |
| " 8 5 \n", | |
| " 3⋅u ⋅Δt \n", | |
| "- ───────────────────────────────────────────────────────────────────────── + \n", | |
| " ⎛ ________ ________ ________ ________⎞ \n", | |
| " ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ \n", | |
| " 16⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " ________ \n", | |
| " 6 5 ╱ 2 6 \n", | |
| " 3844175612059⋅u ⋅Δt ⋅╲╱ 1 - u 3⋅u ⋅\n", | |
| "────────────────────────────────────── + ─────────────────────────────────────\n", | |
| " ⎛ 6 4 2 ⎞ ⎛ ________ _______\n", | |
| "6953849118720⋅⎝- u + 3⋅u - 3⋅u + 1⎠ ⎜ 6 ╱ 2 4 ╱ 2\n", | |
| " 8⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u \n", | |
| "\n", | |
| " \n", | |
| " 5 \n", | |
| "Δt \n", | |
| "─────────────────────────────────── + ────────────────────────────────────────\n", | |
| "_ ________ ________⎞ ________ \n", | |
| " 2 ╱ 2 ╱ 2 ⎟ 4 ╱ 2 \n", | |
| " - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 42730909364772720000⋅u ⋅╲╱ 1 - u - 85\n", | |
| "\n", | |
| " \n", | |
| " 6 5 \n", | |
| " 902475437790052823⋅u ⋅Δt \n", | |
| "──────────────────────────────────────────────────────────────────── + ───────\n", | |
| " ________ ________ \n", | |
| " 2 ╱ 2 ╱ 2 4 \n", | |
| "461818729545440000⋅u ⋅╲╱ 1 - u + 42730909364772720000⋅╲╱ 1 - u 80⋅u ⋅╲\n", | |
| "\n", | |
| " \n", | |
| " 6 5 \n", | |
| " u ⋅Δt 54074\n", | |
| "──────────────────────────────────────────────── - ───────────────────────────\n", | |
| " ________ ________ ________ ⎛ ________\n", | |
| " ╱ 2 2 ╱ 2 ╱ 2 ⎜ 4 ╱ 2 \n", | |
| "╱ 1 - u - 160⋅u ⋅╲╱ 1 - u + 80⋅╲╱ 1 - u 13414060800⋅⎝u ⋅╲╱ 1 - u \n", | |
| "\n", | |
| " ________ \n", | |
| " 6 5 4 5 ╱ 2 \n", | |
| "45509⋅u ⋅Δt 3844175612059⋅u ⋅Δt ⋅╲╱ 1 - u \n", | |
| "────────────────────────────────── - ────────────────────────────────────── - \n", | |
| " ________ ________⎞ ⎛ 6 4 2 ⎞ \n", | |
| " 2 ╱ 2 ╱ 2 ⎟ 6953849118720⋅⎝- u + 3⋅u - 3⋅u + 1⎠ \n", | |
| " - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 3⋅u ⋅Δt \n", | |
| "───────────────────────────────────────────────────────────────────────── - ──\n", | |
| " ⎛ ________ ________ ________ ________⎞ \n", | |
| " ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ \n", | |
| "16⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 42\n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 902475437790052823⋅u ⋅Δt \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| " ________ ________ \n", | |
| " 4 ╱ 2 2 ╱ 2 \n", | |
| "730909364772720000⋅u ⋅╲╱ 1 - u - 85461818729545440000⋅u ⋅╲╱ 1 - u + 4273\n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " u ⋅Δt \n", | |
| "──────────────────────────── - ───────────────────────────────────────────────\n", | |
| " ________ ________ ________ \n", | |
| " ╱ 2 4 ╱ 2 2 ╱ 2 ╱\n", | |
| "0909364772720000⋅╲╱ 1 - u 80⋅u ⋅╲╱ 1 - u - 160⋅u ⋅╲╱ 1 - u + 80⋅╲╱ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 8425609189⋅u ⋅Δt \n", | |
| "──────── + ───────────────────────────────────────────────────────────── + ───\n", | |
| "________ ⎛ ________ ________ ________⎞ \n", | |
| " 2 ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ \n", | |
| " 1 - u 13414060800⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 241\n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 328105563238046242399⋅u ⋅Δt 9⋅\n", | |
| "─────────────────────────────────────────────────── - ────────────────────────\n", | |
| " ⎛ ________ ________⎞ ⎛ ________ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 \n", | |
| "303958765775360000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 40⋅⎝u ⋅╲╱ 1 - u - 2⋅u\n", | |
| "\n", | |
| " \n", | |
| " 2 5 2 5 \n", | |
| "u ⋅Δt 3635043306271280591533⋅u ⋅Δt \n", | |
| "──────────────────────────── - ───────────────────────────────────────────────\n", | |
| " ________ ________⎞ ⎛ ________ _\n", | |
| "2 ╱ 2 ╱ 2 ⎟ ⎜ 2 ╱ 2 ╱ \n", | |
| " ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 4102167299018181120000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ \n", | |
| "\n", | |
| " \n", | |
| " 2 5 5 \n", | |
| " 754418498879⋅u ⋅Δt 3⋅Δt \n", | |
| "──────── + ──────────────────────── - ─────────────────────────────────── - ──\n", | |
| "_______⎞ ________ ⎛ ________ ________⎞ \n", | |
| " 2 ⎟ ╱ 2 ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| "1 - u ⎠ 772649902080⋅╲╱ 1 - u 80⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 69\n", | |
| "\n", | |
| " ⎛ \n", | |
| " 5 ⎛ 5 5 ⎞ ⎜ \n", | |
| " 3844175612059⋅Δt 5 ⎜u⋅Δt Δt ⋅sin(t)⎟ 4 ⎜ \n", | |
| "─────────────────────── + λ ⋅⎜───── - ──────────⎟ + λ ⋅⎜- ────────────────────\n", | |
| " ________ ⎝ 80 80 ⎠ ⎜ ⎛ ________\n", | |
| " ╱ 2 ⎜ ⎜ 6 ╱ 2 \n", | |
| "53849118720⋅╲╱ 1 - u ⎝ 16⋅⎝- u ⋅╲╱ 1 - u \n", | |
| "\n", | |
| " \n", | |
| " 8 5 \n", | |
| " 3⋅u ⋅Δt \n", | |
| "───────────────────────────────────────────────────── + ──────────────────────\n", | |
| " ________ ________ ________⎞ ⎛ ________ \n", | |
| " 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 6 ╱ 2 \n", | |
| " + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 4⋅⎝- u ⋅╲╱ 1 - u + \n", | |
| "\n", | |
| " \n", | |
| " 7 5 \n", | |
| " 3⋅u ⋅Δt ⋅sin(t) \n", | |
| "────────────────────────────────────────────────── - ─────────────────────────\n", | |
| " ________ ________ ________⎞ ⎛ ________ \n", | |
| " 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 6 ╱ 2 \n", | |
| "3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 8⋅⎝- u ⋅╲╱ 1 - u + 3⋅u\n", | |
| "\n", | |
| " \n", | |
| " 6 5 2 \n", | |
| " 9⋅u ⋅Δt ⋅sin (t) 540744\n", | |
| "─────────────────────────────────────────────── - ────────────────────────────\n", | |
| " ________ ________ ________⎞ ⎛ ________ \n", | |
| "4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 \n", | |
| " ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 13414060800⋅⎝u ⋅╲╱ 1 - u \n", | |
| "\n", | |
| " \n", | |
| " 6 5 5 5 3 \n", | |
| "5509⋅u ⋅Δt 3⋅u ⋅Δt ⋅sin (\n", | |
| "───────────────────────────────── + ──────────────────────────────────────────\n", | |
| " ________ ________⎞ ⎛ ________ ________ \n", | |
| " 2 ╱ 2 ╱ 2 ⎟ ⎜ 6 ╱ 2 4 ╱ 2 \n", | |
| "- 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3\n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| "t) 6413500069⋅u ⋅Δt ⋅sin(t) \n", | |
| "────────────────────────────── + ─────────────────────────────────────────────\n", | |
| " ________ ________⎞ ⎛ ________ ________\n", | |
| " 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 \n", | |
| "⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u \n", | |
| "\n", | |
| " \n", | |
| " 4 5 4 \n", | |
| " 3⋅u ⋅Δt ⋅sin (t) \n", | |
| "─────────────── - ────────────────────────────────────────────────────────────\n", | |
| " ________⎞ ⎛ ________ ________ ________ \n", | |
| " ╱ 2 ⎟ ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 \n", | |
| " + ╲╱ 1 - u ⎠ 16⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u +\n", | |
| "\n", | |
| " \n", | |
| " 4 5 2 \n", | |
| " 8425609189⋅u ⋅Δt ⋅sin (t) \n", | |
| "───────────── - ──────────────────────────────────────────────────────────── +\n", | |
| " ________⎞ ⎛ ________ ________ ________⎞ \n", | |
| " ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ \n", | |
| " ╲╱ 1 - u ⎠ 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 357614120019468965321⋅u ⋅Δt 1\n", | |
| " ─────────────────────────────────────────────────────── + ───────────────────\n", | |
| " ⎛ ________ ________⎞ ⎛ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱\n", | |
| " 2051083649509090560000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 13414060800⋅⎝u ⋅╲╱ \n", | |
| "\n", | |
| " \n", | |
| " 3 5 3 \n", | |
| "4461936549⋅u ⋅Δt ⋅sin (t) 98031318961151965421⋅u\n", | |
| "────────────────────────────────────────── - ─────────────────────────────────\n", | |
| "________ ________ ________⎞ ⎛ _\n", | |
| " 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 2 ╱ \n", | |
| " 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 1025541824754545280000⋅⎝- u ⋅╲╱ \n", | |
| "\n", | |
| " \n", | |
| "3 5 2 5 4 \n", | |
| " ⋅Δt ⋅sin(t) 9⋅u ⋅Δt ⋅sin (t) \n", | |
| "────────────────────── - ──────────────────────────────────────────────────── \n", | |
| "_______ ________⎞ ⎛ ________ ________ ________⎞ \n", | |
| " 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ \n", | |
| "1 - u + ╲╱ 1 - u ⎠ 40⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 2 5 2 \n", | |
| " 757632721070389930279⋅u ⋅Δt ⋅sin (t) 697309902804057\n", | |
| "- ─────────────────────────────────────────────────────── + ──────────────────\n", | |
| " ⎛ ________ ________⎞ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| " 2051083649509090560000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 205108364950909056\n", | |
| "\n", | |
| " \n", | |
| " 2 5 5 3 \n", | |
| "541121⋅u ⋅Δt 4401390949⋅u⋅Δt ⋅sin (t) 748586994041\n", | |
| "──────────────── + ──────────────────────────────────────────── - ────────────\n", | |
| " ________ ⎛ ________ ________⎞ \n", | |
| " ╱ 2 ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| "0000⋅╲╱ 1 - u 13414060800⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 102554182475\n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 ╱ 2 5 5 4 \n", | |
| "784805121⋅u⋅Δt ⋅sin(t) Δt ⋅╲╱ 1 - u Δt ⋅cos(t) 3⋅Δt ⋅sin (\n", | |
| "────────────────────── + ─────────────── - ────────── - ──────────────────────\n", | |
| " ________ 80 80 ⎛ ________ \n", | |
| " ╱ 2 ⎜ 2 ╱ 2 \n", | |
| "4545280000⋅╲╱ 1 - u 80⋅⎝- u ⋅╲╱ 1 - u +\n", | |
| "\n", | |
| " ⎞ ⎛ \n", | |
| " 5 2 ⎟ ⎜ \n", | |
| "t) 799864085279512069121⋅Δt ⋅sin (t) ⎟ 3 ⎜ \n", | |
| "───────────── + ──────────────────────────────────⎟ + λ ⋅⎜- ──────────────────\n", | |
| " ________⎞ ________⎟ ⎜ ⎛ _______\n", | |
| " ╱ 2 ⎟ ╱ 2 ⎟ ⎜ ⎜ 6 ╱ 2\n", | |
| " ╲╱ 1 - u ⎠ 2051083649509090560000⋅╲╱ 1 - u ⎠ ⎝ 4⋅⎝- u ⋅╲╱ 1 - u \n", | |
| "\n", | |
| " ________ \n", | |
| " 7 5 ╱ 2 \n", | |
| " 3⋅u ⋅Δt ⋅╲╱ 1 - u 384417561205\n", | |
| "────────────────────────────────────────────────────── - ─────────────────────\n", | |
| "_ ________ ________ ________⎞ ⎛ 6 \n", | |
| " 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 6953849118720⋅⎝- u +\n", | |
| " + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 7 5 7 5 \n", | |
| "9⋅u ⋅Δt 3⋅u ⋅Δt 18\n", | |
| "───────────────── + ────────────────────────────── + ─────────────────────────\n", | |
| " 4 2 ⎞ 6 4 2 6 \n", | |
| " 3⋅u - 3⋅u + 1⎠ - 80⋅u + 240⋅u - 240⋅u + 80 - 13414060800⋅u + 402421\n", | |
| " \n", | |
| "\n", | |
| " __\n", | |
| " 7 5 6 5 ╱ \n", | |
| "86254549⋅u ⋅Δt 9⋅u ⋅Δt ⋅╲╱ 1\n", | |
| "─────────────────────────────────────── + ────────────────────────────────────\n", | |
| " 4 2 ⎛ ________ ______\n", | |
| "82400⋅u - 40242182400⋅u + 13414060800 ⎜ 6 ╱ 2 4 ╱ \n", | |
| " 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u\n", | |
| "\n", | |
| "______ \n", | |
| " 2 6 5 \n", | |
| " - u ⋅sin(t) 3844175612059⋅u ⋅Δt ⋅sin(t) \n", | |
| "──────────────────────────────────── + ────────────────────────────────────── \n", | |
| "__ ________ ________⎞ ⎛ 6 4 2 ⎞ \n", | |
| "2 2 ╱ 2 ╱ 2 ⎟ 2317949706240⋅⎝- u + 3⋅u - 3⋅u + 1⎠ \n", | |
| " - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 6 5 6 5 \n", | |
| " 9⋅u ⋅Δt ⋅sin(t) 5658763647⋅u ⋅Δt ⋅sin(t\n", | |
| "- ────────────────────────────── - ───────────────────────────────────────────\n", | |
| " 6 4 2 6 4 \n", | |
| " - 80⋅u + 240⋅u - 240⋅u + 80 - 13414060800⋅u + 40242182400⋅u - 4024218\n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 ╱ 2 2 \n", | |
| ") 9⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin (t) \n", | |
| "───────────────────── - ──────────────────────────────────────────────────────\n", | |
| " 2 ⎛ ________ ________ _____\n", | |
| "2400⋅u + 13414060800 ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ \n", | |
| " 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 ╱ 2 \n", | |
| " 6413500069⋅u ⋅Δt ⋅╲╱ 1 - u \n", | |
| "────────────────── - ─────────────────────────────────────────────────────────\n", | |
| "___ ________⎞ ⎛ ________ ________ ______\n", | |
| " 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| "u + ╲╱ 1 - u ⎠ 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u\n", | |
| "\n", | |
| " \n", | |
| " 5 5 2 5 5 2 \n", | |
| " 3844175612059⋅u ⋅Δt ⋅sin (t) 9⋅u ⋅Δt ⋅sin (t) \n", | |
| "─── - ────────────────────────────────────── + ────────────────────────────── \n", | |
| "__⎞ ⎛ 6 4 2 ⎞ 6 4 2 \n", | |
| "2 ⎟ 2317949706240⋅⎝- u + 3⋅u - 3⋅u + 1⎠ - 80⋅u + 240⋅u - 240⋅u + 80 \n", | |
| " ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 5 5 2 \n", | |
| " 5658763647⋅u ⋅Δt ⋅sin (t) \n", | |
| "+ ──────────────────────────────────────────────────────────────── + ─────────\n", | |
| " 6 4 2 \n", | |
| " - 13414060800⋅u + 40242182400⋅u - 40242182400⋅u + 13414060800 \n", | |
| " 134140608\n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 1886254549⋅u ⋅Δt ⋅cos(t) \n", | |
| "──────────────────────────────────────────────────── + ───────────────────────\n", | |
| " ⎛ ________ ________ ________⎞ 4\n", | |
| " ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 21365454682386360000⋅u \n", | |
| "00⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| "902475437790052823⋅u ⋅Δt 1634167659\n", | |
| "───────────────────────────────────────────────── - ──────────────────────────\n", | |
| " 2 4 \n", | |
| " - 42730909364772720000⋅u + 21365454682386360000 17384622796800⋅u - 347692\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 5 5 5 5 \n", | |
| "4871⋅u ⋅Δt 105199674209⋅u ⋅Δt \n", | |
| "──────────────────────────── - ───────────────────────────────────────────────\n", | |
| " 2 4 2 \n", | |
| "45593600⋅u + 17384622796800 124175877120⋅u - 248351754240⋅u + 12417587712\n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 4 5 ╱ 2 3 \n", | |
| " 3⋅u ⋅Δt 3⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin (t) \n", | |
| "─ + ─────────────────── + ────────────────────────────────────────────────────\n", | |
| " 4 2 ⎛ ________ ________ ___\n", | |
| "0 80⋅u - 160⋅u + 80 ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ \n", | |
| " 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 \n", | |
| "\n", | |
| " ________ \n", | |
| " 4 5 ╱ 2 \n", | |
| " 8425609189⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) \n", | |
| "──────────────────── + ───────────────────────────────────────────────────────\n", | |
| "_____ ________⎞ ⎛ ________ ________ ____\n", | |
| " 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| "- u + ╲╱ 1 - u ⎠ 2235676800⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 -\n", | |
| "\n", | |
| " \n", | |
| " 4 5 3 4 5 3 \n", | |
| " 3844175612059⋅u ⋅Δt ⋅sin (t) 3⋅u ⋅Δt ⋅sin (t) \n", | |
| "───── + ────────────────────────────────────── - ─────────────────────────────\n", | |
| "____⎞ ⎛ 6 4 2 ⎞ 6 4 2 \n", | |
| " 2 ⎟ 6953849118720⋅⎝- u + 3⋅u - 3⋅u + 1⎠ - 80⋅u + 240⋅u - 240⋅u + 8\n", | |
| " u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 3 \n", | |
| " 1886254549⋅u ⋅Δt ⋅sin (t) \n", | |
| "─ - ──────────────────────────────────────────────────────────────── - ───────\n", | |
| " 6 4 2 \n", | |
| "0 - 13414060800⋅u + 40242182400⋅u - 40242182400⋅u + 13414060800 \n", | |
| " 6707030\n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 1886254549⋅u ⋅Δt ⋅sin(t)⋅cos(t) 90\n", | |
| "───────────────────────────────────────────────────── - ──────────────────────\n", | |
| " ⎛ ________ ________ ________⎞ \n", | |
| " ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 10682727341193180000⋅u\n", | |
| "400⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| "2475437790052823⋅u ⋅Δt ⋅sin(t) 494596453545\n", | |
| "────────────────────────────────────────────────── + ─────────────────────────\n", | |
| "4 2 4 \n", | |
| " - 21365454682386360000⋅u + 10682727341193180000 17384622796800⋅u - 34769\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 4 5 4 5 \n", | |
| "33⋅u ⋅Δt ⋅sin(t) 105199674209⋅u ⋅Δt ⋅sin(t) \n", | |
| "───────────────────────────── + ──────────────────────────────────────────────\n", | |
| " 2 4 2 \n", | |
| "245593600⋅u + 17384622796800 62087938560⋅u - 124175877120⋅u + 62087938560\n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 4 5 3 5 ╱ 2 2 \n", | |
| " 9⋅u ⋅Δt ⋅sin(t) 14461936549⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin (t) \n", | |
| " - ─────────────────── - ─────────────────────────────────────────────────────\n", | |
| " 4 2 ⎛ ________ ________ __\n", | |
| " 80⋅u - 160⋅u + 80 ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| " 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1\n", | |
| "\n", | |
| " ________ \n", | |
| " 3 5 ╱ 2 \n", | |
| " 98031318961151965421⋅u ⋅Δt ⋅╲╱ 1 - u \n", | |
| "─────── + ─────────────────────────────────────────────────────── + ──────────\n", | |
| "______⎞ ⎛ ________ ________⎞ \n", | |
| " 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| " - u ⎠ 1025541824754545280000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 1341406080\n", | |
| "\n", | |
| " \n", | |
| " 3 5 2 \n", | |
| " 1886254549⋅u ⋅Δt ⋅sin (t)⋅cos(t) 90247\n", | |
| "─────────────────────────────────────────────────── + ────────────────────────\n", | |
| " ⎛ ________ ________ ________⎞ 4 \n", | |
| " ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 21365454682386360000⋅u \n", | |
| "0⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 3 5 2 \n", | |
| "5437790052823⋅u ⋅Δt ⋅sin (t) 49894260924453⋅\n", | |
| "──────────────────────────────────────────────── - ───────────────────────────\n", | |
| " 2 4 \n", | |
| "- 42730909364772720000⋅u + 21365454682386360000 17384622796800⋅u - 3476924\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 3 5 2 3 5 2 \n", | |
| "u ⋅Δt ⋅sin (t) 105199674209⋅u ⋅Δt ⋅sin (t) \n", | |
| "─────────────────────────── - ────────────────────────────────────────────────\n", | |
| " 2 4 2 \n", | |
| "5593600⋅u + 17384622796800 124175877120⋅u - 248351754240⋅u + 124175877120\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 3 5 2 3 5 \n", | |
| " 9⋅u ⋅Δt ⋅sin (t) 493765118337591853817⋅u ⋅Δt ⋅cos(t) \n", | |
| " + ─────────────────── - ─────────────────────────────────────────────────────\n", | |
| " 4 2 ⎛ ________ _______\n", | |
| " 80⋅u - 160⋅u + 80 ⎜ 2 ╱ 2 ╱ 2\n", | |
| " 2051083649509090560000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u \n", | |
| "\n", | |
| " ________ \n", | |
| " 3 5 2 5 ╱ 2 3 \n", | |
| " 10319047364622345641093⋅u ⋅Δt 9⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin (t\n", | |
| "── - ─────────────────────────────── + ───────────────────────────────────────\n", | |
| "_⎞ ⎛ 2⎞ ⎛ ________ ________ \n", | |
| " ⎟ 8204334598036362240000⋅⎝1 - u ⎠ ⎜ 4 ╱ 2 2 ╱ 2 \n", | |
| " ⎠ 10⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u +\n", | |
| "\n", | |
| " ________ \n", | |
| " 2 5 ╱ 2 \n", | |
| ") 757632721070389930279⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) \n", | |
| "───────────── + ─────────────────────────────────────────────────────── + ────\n", | |
| " ________⎞ ⎛ ________ ________⎞ \n", | |
| " ╱ 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ 1738\n", | |
| " ╲╱ 1 - u ⎠ 1025541824754545280000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 2 5 3 2 5 3 \n", | |
| " 16776292164791⋅u ⋅Δt ⋅sin (t) 3⋅u ⋅Δt ⋅sin (t) \n", | |
| "────────────────────────────────────────────────── - ─────────────────── + ───\n", | |
| " 4 2 4 2 \n", | |
| "4622796800⋅u - 34769245593600⋅u + 17384622796800 80⋅u - 160⋅u + 80 \n", | |
| " 120\n", | |
| "\n", | |
| " \n", | |
| " 2 5 \n", | |
| " 12079201575925326001⋅u ⋅Δt ⋅sin(t)⋅cos(t) 25788879601336394706311⋅\n", | |
| "─────────────────────────────────────────────────── + ────────────────────────\n", | |
| " ⎛ ________ ________⎞ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ 820433459803636224000\n", | |
| "651979382887680000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " ________ \n", | |
| " 2 5 5 ╱ 2 2 \n", | |
| "u ⋅Δt ⋅sin(t) 4401390949⋅u⋅Δt ⋅╲╱ 1 - u ⋅sin (t) 7485869940417848\n", | |
| "───────────── - ─────────────────────────────────────────── + ────────────────\n", | |
| " ⎛ 2⎞ ⎛ ________ ________⎞ 1025541824754\n", | |
| "0⋅⎝1 - u ⎠ ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| " 4471353600⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 5 5 2 \n", | |
| "05121⋅u⋅Δt 1886254549⋅u⋅Δt ⋅sin (t)⋅cos(t) 20005292013953025\n", | |
| "─────────── + ──────────────────────────────────────────── - ─────────────────\n", | |
| "545280000 ⎛ ________ ________⎞ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ 82043345980363\n", | |
| " 13414060800⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " ____\n", | |
| " 5 2 5 5 ╱ \n", | |
| "321343⋅u⋅Δt ⋅sin (t) 705268173027862269617⋅u⋅Δt ⋅cos(t) 3⋅Δt ⋅╲╱ 1 -\n", | |
| "──────────────────── - ────────────────────────────────── + ──────────────────\n", | |
| " ⎛ 2⎞ ________ ⎛ ______\n", | |
| "62240000⋅⎝1 - u ⎠ ╱ 2 ⎜ 2 ╱ \n", | |
| " 2051083649509090560000⋅╲╱ 1 - u 20⋅⎝- u ⋅╲╱ 1 - u\n", | |
| "\n", | |
| "____ \n", | |
| " 2 3 5 5 3 \n", | |
| " u ⋅sin (t) 787044812470080253121⋅Δt ⋅sin(t) 3844175612059⋅Δt ⋅sin (\n", | |
| "───────────────── - ──────────────────────────────── + ───────────────────────\n", | |
| "__ ________⎞ 1025541824754545280000 ⎛ 2\n", | |
| "2 ╱ 2 ⎟ 6953849118720⋅⎝1 - u \n", | |
| " + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " ⎞ ⎛ \n", | |
| " 5 ⎟ ⎜ \n", | |
| "t) 782183809884453165617⋅Δt ⋅sin(t)⋅cos(t)⎟ 2 ⎜ \n", | |
| "── + ───────────────────────────────────────⎟ + λ ⋅⎜──────────────────────────\n", | |
| "⎞ ________ ⎟ ⎜ ⎛ ________ \n", | |
| "⎠ ╱ 2 ⎟ ⎜ ⎜ 6 ╱ 2 4\n", | |
| " 2051083649509090560000⋅╲╱ 1 - u ⎠ ⎝8⋅⎝- u ⋅╲╱ 1 - u + 3⋅u \n", | |
| "\n", | |
| " \n", | |
| " 8 5 \n", | |
| " 9⋅u ⋅Δt 9\n", | |
| "────────────────────────────────────────────── - ─────────────────────────────\n", | |
| " ________ ________ ________⎞ ⎛ ________ \n", | |
| " ╱ 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 6 ╱ 2 4 \n", | |
| "⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱\n", | |
| "\n", | |
| " _____\n", | |
| " 7 5 6 5 ╱ \n", | |
| "⋅u ⋅Δt ⋅sin(t) 3844175612059⋅u ⋅Δt ⋅╲╱ 1 - \n", | |
| "─────────────────────────────────────────── - ────────────────────────────────\n", | |
| " ________ ________ ________⎞ ⎛ 6 4 \n", | |
| "╱ 2 2 ╱ 2 ╱ 2 ⎟ 2317949706240⋅⎝- u + 3⋅u - 3⋅u\n", | |
| " 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| "___ \n", | |
| " 2 6 5 2 \n", | |
| "u 9⋅u ⋅Δt ⋅sin (t) \n", | |
| "────── + ─────────────────────────────────────────────────────────────────────\n", | |
| "2 ⎞ ⎛ ________ ________ ________ ______\n", | |
| " + 1⎠ ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| " 8⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u\n", | |
| "\n", | |
| " \n", | |
| " 6 5 \n", | |
| " 9⋅u ⋅Δt \n", | |
| "─── - ────────────────────────────────────────────────────────────────────────\n", | |
| "__⎞ ⎛ ________ ________ ________ ________⎞\n", | |
| "2 ⎟ ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟\n", | |
| " ⎠ 8⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠\n", | |
| "\n", | |
| " \n", | |
| " 6 5 \n", | |
| " 902475437790052823⋅u ⋅Δt \n", | |
| " - ───────────────────────────────────────────────────────────────────────────\n", | |
| " ________ ________ \n", | |
| " 4 ╱ 2 2 ╱ 2 \n", | |
| " 42730909364772720000⋅u ⋅╲╱ 1 - u - 85461818729545440000⋅u ⋅╲╱ 1 - u +\n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " 1051996742\n", | |
| "───────────────────────────────── + ──────────────────────────────────────────\n", | |
| " ________ ________ \n", | |
| " ╱ 2 4 ╱ 2 \n", | |
| " 42730909364772720000⋅╲╱ 1 - u 248351754240⋅u ⋅╲╱ 1 - u - 496703508480\n", | |
| "\n", | |
| " \n", | |
| " 6 5 6 5 \n", | |
| "09⋅u ⋅Δt u ⋅Δt \n", | |
| "────────────────────────────────────────── - ─────────────────────────────────\n", | |
| " ________ ________ ________ ___\n", | |
| " 2 ╱ 2 ╱ 2 4 ╱ 2 2 ╱ \n", | |
| "⋅u ⋅╲╱ 1 - u + 248351754240⋅╲╱ 1 - u 80⋅u ⋅╲╱ 1 - u - 160⋅u ⋅╲╱ 1 \n", | |
| "\n", | |
| " \n", | |
| " 6 5 \n", | |
| " 5407445509⋅u ⋅Δt \n", | |
| "────────────────────── + ─────────────────────────────────────────────────────\n", | |
| "_____ ________ ⎛ ________ ________ __\n", | |
| " 2 ╱ 2 ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| "- u + 80⋅╲╱ 1 - u 2235676800⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1\n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 ╱ 2 \n", | |
| " 3844175612059⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) \n", | |
| "─────── + ─────────────────────────────────────── + ──────────────────────────\n", | |
| "______⎞ ⎛ 6 4 2 ⎞ ⎛ ________ \n", | |
| " 2 ⎟ 1158974853120⋅⎝- u + 3⋅u - 3⋅u + 1⎠ ⎜ 6 ╱ 2 4\n", | |
| " - u ⎠ 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 9⋅u ⋅Δt ⋅sin(t) \n", | |
| "────────────────────────────────────────────── + ─────────────────────────────\n", | |
| " ________ ________ ________⎞ __\n", | |
| " ╱ 2 2 ╱ 2 ╱ 2 ⎟ 4 ╱ \n", | |
| "⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 21365454682386360000⋅u ⋅╲╱ 1\n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 902475437790052823⋅u ⋅Δt ⋅sin(t) \n", | |
| "──────────────────────────────────────────────────────────────────────────────\n", | |
| "______ ________ _______\n", | |
| " 2 2 ╱ 2 ╱ 2\n", | |
| " - u - 42730909364772720000⋅u ⋅╲╱ 1 - u + 21365454682386360000⋅╲╱ 1 - u \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 105199674209⋅u ⋅Δt ⋅sin(t) \n", | |
| "─ - ──────────────────────────────────────────────────────────────────────────\n", | |
| "_ ________ ________ \n", | |
| " 4 ╱ 2 2 ╱ 2 \n", | |
| " 124175877120⋅u ⋅╲╱ 1 - u - 248351754240⋅u ⋅╲╱ 1 - u + 124175877120⋅╲\n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 2⋅u ⋅Δt ⋅sin(t) \n", | |
| "────────── + ─────────────────────────────────────────────────────── - ───────\n", | |
| " ________ ________ ________ ________ \n", | |
| " ╱ 2 4 ╱ 2 2 ╱ 2 ╱ 2 \n", | |
| "╱ 1 - u 80⋅u ⋅╲╱ 1 - u - 160⋅u ⋅╲╱ 1 - u + 80⋅╲╱ 1 - u 1490451\n", | |
| "\n", | |
| " \n", | |
| " 5 5 4 5 \n", | |
| " 6413500069⋅u ⋅Δt ⋅sin(t) 3844175612059⋅u ⋅Δt ⋅╲\n", | |
| "───────────────────────────────────────────────────── - ──────────────────────\n", | |
| " ⎛ ________ ________ ________⎞ ⎛ 6 \n", | |
| " ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 2317949706240⋅⎝- u +\n", | |
| "200⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " ________ ________ \n", | |
| " ╱ 2 2 4 5 ╱ 2 \n", | |
| "╱ 1 - u ⋅sin (t) 1886254549⋅u ⋅Δt ⋅╲╱ 1 - u ⋅cos(t) \n", | |
| "────────────────── + ─────────────────────────────────────────────────────────\n", | |
| " 4 2 ⎞ ⎛ ________ ________ ______\n", | |
| " 3⋅u - 3⋅u + 1⎠ ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| " 6707030400⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u\n", | |
| "\n", | |
| " \n", | |
| " 4 5 2 \n", | |
| " 9⋅u ⋅Δt ⋅sin (t) \n", | |
| "─── - ────────────────────────────────────────────────────────────────────────\n", | |
| "__⎞ ⎛ ________ ________ ________ ________⎞\n", | |
| "2 ⎟ ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟\n", | |
| " ⎠ 8⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠\n", | |
| "\n", | |
| " \n", | |
| " 4 5 2 \n", | |
| " 902475437790052823⋅u ⋅Δt ⋅sin (t) \n", | |
| " - ───────────────────────────────────────────────────────────────────────────\n", | |
| " ________ ________ \n", | |
| " 4 ╱ 2 2 ╱ 2 \n", | |
| " 42730909364772720000⋅u ⋅╲╱ 1 - u - 85461818729545440000⋅u ⋅╲╱ 1 - u +\n", | |
| "\n", | |
| " \n", | |
| " \n", | |
| " 105199674209⋅u\n", | |
| "───────────────────────────────── + ──────────────────────────────────────────\n", | |
| " ________ ________ \n", | |
| " ╱ 2 4 ╱ 2 \n", | |
| " 42730909364772720000⋅╲╱ 1 - u 248351754240⋅u ⋅╲╱ 1 - u - 496703508480\n", | |
| "\n", | |
| " \n", | |
| "4 5 2 4 5 2 \n", | |
| " ⋅Δt ⋅sin (t) u ⋅Δt ⋅sin (\n", | |
| "────────────────────────────────────────── - ─────────────────────────────────\n", | |
| " ________ ________ ________ ___\n", | |
| " 2 ╱ 2 ╱ 2 4 ╱ 2 2 ╱ \n", | |
| "⋅u ⋅╲╱ 1 - u + 248351754240⋅╲╱ 1 - u 80⋅u ⋅╲╱ 1 - u - 160⋅u ⋅╲╱ 1 \n", | |
| "\n", | |
| " \n", | |
| " 4 5 2 \n", | |
| "t) 8425609189⋅u ⋅Δt ⋅sin (t) \n", | |
| "────────────────────── + ─────────────────────────────────────────────────────\n", | |
| "_____ ________ ⎛ ________ ________ __\n", | |
| " 2 ╱ 2 ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| "- u + 80⋅╲╱ 1 - u 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1\n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 8425609189⋅u ⋅Δt \n", | |
| "─────── - ──────────────────────────────────────────────────────────── + ─────\n", | |
| "______⎞ ⎛ ________ ________ ________⎞ \n", | |
| " 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 20510\n", | |
| " - u ⎠ 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| " 782183809884453165617⋅u ⋅Δt ⋅cos(t) \n", | |
| "───────────────────────────────────────────────────────────────────────── - ──\n", | |
| " 4 2 \n", | |
| "83649509090560000⋅u - 4102167299018181120000⋅u + 2051083649509090560000 \n", | |
| " 41\n", | |
| "\n", | |
| " \n", | |
| " 4 5 3 \n", | |
| " 14903400795783417131743⋅u ⋅Δt 1886254549⋅u ⋅\n", | |
| "───────────────────────────────────────────────────── - ──────────────────────\n", | |
| " ⎛ ________ ________⎞ ⎛ ____\n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ \n", | |
| "02167299018181120000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 6707030400⋅⎝u ⋅╲╱ 1 -\n", | |
| "\n", | |
| " ________ \n", | |
| " 5 ╱ 2 3 5 \n", | |
| "Δt ⋅╲╱ 1 - u ⋅sin(t)⋅cos(t) 14461936549⋅u ⋅Δt ⋅s\n", | |
| "────────────────────────────────────── + ─────────────────────────────────────\n", | |
| "____ ________ ________⎞ ⎛ ________ \n", | |
| " 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱\n", | |
| " u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ \n", | |
| "\n", | |
| " \n", | |
| " 3 5 \n", | |
| "in(t) 782183809884453165617⋅u ⋅Δt ⋅sin(t\n", | |
| "─────────────────────── - ────────────────────────────────────────────────────\n", | |
| "________ ________⎞ 4 \n", | |
| " 2 ╱ 2 ⎟ 2051083649509090560000⋅u - 4102167299018181120000⋅u\n", | |
| " 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 3 5 \n", | |
| ")⋅cos(t) 48926120124603963505793⋅u ⋅Δt ⋅sin(t) \n", | |
| "────────────────────────── + ─────────────────────────────────────────────────\n", | |
| "2 ⎛ ________ ___\n", | |
| " + 2051083649509090560000 ⎜ 2 ╱ 2 ╱ \n", | |
| " 8204334598036362240000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 \n", | |
| "\n", | |
| " ________ \n", | |
| " 2 5 ╱ 2 \n", | |
| " 12079201575925326001⋅u ⋅Δt ⋅╲╱ 1 - u ⋅cos(t) \n", | |
| "────── - ────────────────────────────────────────────────────── - ────────────\n", | |
| "_____⎞ ⎛ ________ ________⎞ ⎛ __\n", | |
| " 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ \n", | |
| "- u ⎠ 120651979382887680000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 20⋅⎝u ⋅╲╱ 1\n", | |
| "\n", | |
| " \n", | |
| " 2 5 2 2\n", | |
| " 27⋅u ⋅Δt ⋅sin (t) 22318774874155177518791⋅u \n", | |
| "──────────────────────────────────────── - ───────────────────────────────────\n", | |
| "______ ________ ________⎞ ⎛ ___\n", | |
| " 2 2 ╱ 2 ╱ 2 ⎟ ⎜ 2 ╱ \n", | |
| " - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 8204334598036362240000⋅⎝- u ⋅╲╱ 1 \n", | |
| "\n", | |
| " \n", | |
| " 5 2 2 5 2 \n", | |
| "⋅Δt ⋅sin (t) 902475437790052823⋅u ⋅Δt ⋅cos (t) \n", | |
| "──────────────────── - ───────────────────────────────────────────────────── -\n", | |
| "_____ ________⎞ ⎛ ________ ________⎞ \n", | |
| " 2 ╱ 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| "- u + ╲╱ 1 - u ⎠ 42730909364772720000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 2 5 \n", | |
| " 757632721070389930279⋅u ⋅Δt 7052681730278622696\n", | |
| " ─────────────────────────────────────────────────────── + ───────────────────\n", | |
| " ⎛ ________ ________⎞ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ 20510836495090905\n", | |
| " 2051083649509090560000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " _\n", | |
| " 2 5 2 5 5 ╱ \n", | |
| "17⋅u ⋅Δt ⋅cos(t) 5102610026961050007259⋅u ⋅Δt 1886254549⋅u⋅Δt ⋅╲╱ \n", | |
| "──────────────── - ────────────────────────────────── - ──────────────────────\n", | |
| " ⎛ 2⎞ ________ ⎛ __\n", | |
| "60000⋅⎝1 - u ⎠ ╱ 2 ⎜ 2 ╱ \n", | |
| " 1640866919607272448000⋅╲╱ 1 - u 6707030400⋅⎝- u ⋅╲╱ 1\n", | |
| "\n", | |
| "_______ \n", | |
| " 2 5 \n", | |
| "1 - u ⋅sin(t)⋅cos(t) 4401390949⋅u⋅Δt ⋅sin(t) 78218380\n", | |
| "───────────────────── + ─────────────────────────────────────────── - ────────\n", | |
| "______ ________⎞ ⎛ ________ ________⎞ \n", | |
| " 2 ╱ 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ 205\n", | |
| " - u + ╲╱ 1 - u ⎠ 4471353600⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| "9884453165617⋅u⋅Δt ⋅sin(t)⋅cos(t) 14780311813398039280937⋅u⋅Δt ⋅sin(t) 105\n", | |
| "───────────────────────────────── + ──────────────────────────────────── + ───\n", | |
| " ⎛ 2⎞ ________ \n", | |
| "1083649509090560000⋅⎝1 - u ⎠ ╱ 2 \n", | |
| " 2734778199345454080000⋅╲╱ 1 - u \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 ╱ 2 5 \n", | |
| "199674209⋅Δt ⋅╲╱ 1 - u 756545264265589533617⋅Δt ⋅cos(t) 9⋅Δt\n", | |
| "───────────────────────── - ──────────────────────────────── - ───────────────\n", | |
| " 248351754240 2051083649509090560000 ⎛ ___\n", | |
| " ⎜ 2 ╱ \n", | |
| " 40⋅⎝- u ⋅╲╱ 1 \n", | |
| "\n", | |
| " \n", | |
| "5 2 5 2 5 \n", | |
| " ⋅sin (t) 3844175612059⋅Δt ⋅sin (t) 902475437790052823⋅Δt ⋅cos\n", | |
| "──────────────────── - ───────────────────────── - ───────────────────────────\n", | |
| "_____ ________⎞ ________ ___\n", | |
| " 2 ╱ 2 ⎟ ╱ 2 ╱ \n", | |
| "- u + ╲╱ 1 - u ⎠ 1738462279680⋅╲╱ 1 - u 42730909364772720000⋅╲╱ 1 \n", | |
| "\n", | |
| " ⎞ ⎛ \n", | |
| "2 5 ⎟ ⎜ 7 \n", | |
| " (t) 1436611804849711823⋅Δt ⎟ ⎜ 3⋅u ⋅\n", | |
| "───── - ────────────────────────────────⎟ + λ⋅⎜───────────────────────────────\n", | |
| "_____ ________⎟ ⎜ ⎛ ________ _\n", | |
| " 2 ╱ 2 ⎟ ⎜ ⎜ 6 ╱ 2 4 ╱ \n", | |
| "- u 42730909364772720000⋅╲╱ 1 - u ⎠ ⎝4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 ╱ 2 7 5 \n", | |
| "Δt ⋅╲╱ 1 - u 3844175612059⋅u ⋅Δt \n", | |
| "───────────────────────────────────────── + ──────────────────────────────────\n", | |
| "_______ ________ ________⎞ ⎛ 6 4 2 \n", | |
| " 2 2 ╱ 2 ╱ 2 ⎟ 2317949706240⋅⎝- u + 3⋅u - 3⋅u \n", | |
| "1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 7 5 \n", | |
| " 3395336389⋅u ⋅Δt \n", | |
| "──── - ──────────────────────────────────────────────────────────────── - ────\n", | |
| " ⎞ 6 4 2 ⎛ \n", | |
| "+ 1⎠ - 13414060800⋅u + 40242182400⋅u - 40242182400⋅u + 13414060800 ⎜ \n", | |
| " 4⋅⎝-\n", | |
| "\n", | |
| " ________ \n", | |
| " 6 5 ╱ 2 \n", | |
| " 3⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) 38\n", | |
| "──────────────────────────────────────────────────────────────────── - ───────\n", | |
| " ________ ________ ________ ________⎞ \n", | |
| " 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 2317949\n", | |
| " u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 6 5 6 5 \n", | |
| "44175612059⋅u ⋅Δt ⋅sin(t) 3395336389⋅u ⋅Δt ⋅sin(t)\n", | |
| "─────────────────────────────── + ────────────────────────────────────────────\n", | |
| " ⎛ 6 4 2 ⎞ 6 4 \n", | |
| "706240⋅⎝- u + 3⋅u - 3⋅u + 1⎠ - 13414060800⋅u + 40242182400⋅u - 40242182\n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 ╱ 2 \n", | |
| " 3⋅u ⋅Δt ⋅╲╱ 1 - u \n", | |
| "──────────────────── - ───────────────────────────────────────────────────────\n", | |
| " 2 ⎛ ________ ________ ______\n", | |
| "400⋅u + 13414060800 ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ \n", | |
| " 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u\n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 ╱ 2 \n", | |
| " 6078148549⋅u ⋅Δt ⋅╲╱ 1 - u \n", | |
| "───────────────── + ──────────────────────────────────────────────────────────\n", | |
| "__ ________⎞ ⎛ ________ ________ _______\n", | |
| "2 ╱ 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2\n", | |
| " + ╲╱ 1 - u ⎠ 4471353600⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 3844175612059⋅u ⋅Δt 339533638\n", | |
| "── - ────────────────────────────────────── + ────────────────────────────────\n", | |
| "_⎞ ⎛ 6 4 2 ⎞ 6 \n", | |
| " ⎟ 2317949706240⋅⎝- u + 3⋅u - 3⋅u + 1⎠ - 13414060800⋅u + 40242182400⋅u\n", | |
| " ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 5 5 5 5 \n", | |
| "9⋅u ⋅Δt 1886254549⋅u ⋅Δt ⋅cos(t)\n", | |
| "──────────────────────────────── - ───────────────────────────────────────────\n", | |
| "4 2 ⎛ ________ _____\n", | |
| " - 40242182400⋅u + 13414060800 ⎜ 4 ╱ 2 2 ╱ \n", | |
| " 13414060800⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 902475437790052823⋅u ⋅Δt \n", | |
| "────────────────── - ─────────────────────────────────────────────────────────\n", | |
| "___ ________⎞ 4 2 \n", | |
| " 2 ╱ 2 ⎟ 10682727341193180000⋅u - 21365454682386360000⋅u + 10682\n", | |
| "u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| " 3186582816619⋅u ⋅Δt \n", | |
| "─────────────── + ─────────────────────────────────────────────────── + ──────\n", | |
| " 4 2 ⎛ 4\n", | |
| "727341193180000 1931624755200⋅u - 3863249510400⋅u + 1931624755200 80⋅⎝u \n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 5 4 5 ╱ 2 \n", | |
| "u ⋅Δt 3⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) \n", | |
| "──────────── + ───────────────────────────────────────────────────────────────\n", | |
| " 2 ⎞ ⎛ ________ ________ ________ \n", | |
| " - 2⋅u + 1⎠ ⎜ 6 ╱ 2 4 ╱ 2 2 ╱ 2 ╱\n", | |
| " 4⋅⎝- u ⋅╲╱ 1 - u + 3⋅u ⋅╲╱ 1 - u - 3⋅u ⋅╲╱ 1 - u + ╲╱ \n", | |
| "\n", | |
| " ________ \n", | |
| " 4 5 ╱ 2 \n", | |
| " 7922581909⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) \n", | |
| "───────── - ──────────────────────────────────────────────────────────── + ───\n", | |
| "________⎞ ⎛ ________ ________ ________⎞ \n", | |
| " 2 ⎟ ⎜ 4 ╱ 2 2 ╱ 2 ╱ 2 ⎟ 231\n", | |
| " 1 - u ⎠ 6707030400⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 4 5 4 5 \n", | |
| " 3844175612059⋅u ⋅Δt ⋅sin(t) 3395336389⋅u ⋅Δt ⋅si\n", | |
| "─────────────────────────────────── - ────────────────────────────────────────\n", | |
| " ⎛ 6 4 2 ⎞ 6 4 \n", | |
| "7949706240⋅⎝- u + 3⋅u - 3⋅u + 1⎠ - 13414060800⋅u + 40242182400⋅u - 4024\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| "n(t) 902475437790052823⋅u ⋅Δt ⋅sin(t\n", | |
| "──────────────────────── + ───────────────────────────────────────────────────\n", | |
| " 2 4 2 \n", | |
| "2182400⋅u + 13414060800 21365454682386360000⋅u - 42730909364772720000⋅u +\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 4 5 \n", | |
| ") 3234873435499⋅u ⋅Δt ⋅sin(t) \n", | |
| "───────────────────── - ─────────────────────────────────────────────────── - \n", | |
| " 4 2 \n", | |
| " 21365454682386360000 1931624755200⋅u - 3863249510400⋅u + 1931624755200 \n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 4 5 3 5 ╱ 2 \n", | |
| " u ⋅Δt ⋅sin(t) 14461936549⋅u ⋅Δt ⋅╲╱ 1 - u \n", | |
| "────────────────── - ─────────────────────────────────────────────────────────\n", | |
| " ⎛ 4 2 ⎞ ⎛ ________ ________ _____\n", | |
| "80⋅⎝u - 2⋅u + 1⎠ ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| " 13414060800⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - \n", | |
| "\n", | |
| " ________ \n", | |
| " 3 5 ╱ 2 \n", | |
| " 22917976411669⋅u ⋅Δt ⋅╲╱ 1 - u 188\n", | |
| "──── - ────────────────────────────────────────────── + ──────────────────────\n", | |
| "___⎞ ⎛ ________ ________⎞ ⎛ ___\n", | |
| " 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ ⎜ 4 ╱ \n", | |
| "u ⎠ 8692311398400⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 13414060800⋅⎝u ⋅╲╱ 1 \n", | |
| "\n", | |
| " \n", | |
| " 3 5 \n", | |
| "6254549⋅u ⋅Δt ⋅cos(t) 9024754377900\n", | |
| "─────────────────────────────────────── + ────────────────────────────────────\n", | |
| "_____ ________ ________⎞ 4 \n", | |
| " 2 2 ╱ 2 ╱ 2 ⎟ 21365454682386360000⋅u - 4273090936\n", | |
| "- u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 3 5 3 5 \n", | |
| "52823⋅u ⋅Δt 14385906530231⋅u ⋅Δt \n", | |
| "──────────────────────────────────── - ───────────────────────────────────────\n", | |
| " 2 4 2 \n", | |
| "4772720000⋅u + 21365454682386360000 17384622796800⋅u - 34769245593600⋅u +\n", | |
| " \n", | |
| "\n", | |
| " \n", | |
| " 3 5 \n", | |
| " 519403663956455485817⋅u ⋅Δt ⋅cos(t) 9\n", | |
| "─────────────── + ─────────────────────────────────────────────────────── + ──\n", | |
| " ⎛ ________ ________⎞ \n", | |
| " 17384622796800 ⎜ 2 ╱ 2 ╱ 2 ⎟ 82\n", | |
| " 2051083649509090560000⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " ________ \n", | |
| " 3 5 2 5 ╱ 2 \n", | |
| "972496796510965357061⋅u ⋅Δt 9⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) \n", | |
| "───────────────────────────── + ──────────────────────────────────────────────\n", | |
| " ⎛ 2⎞ ⎛ ________ ________ ___\n", | |
| "04334598036362240000⋅⎝1 - u ⎠ ⎜ 4 ╱ 2 2 ╱ 2 ╱ \n", | |
| " 10⋅⎝u ⋅╲╱ 1 - u - 2⋅u ⋅╲╱ 1 - u + ╲╱ 1 \n", | |
| "\n", | |
| " ________ \n", | |
| " 2 5 ╱ 2 \n", | |
| " 86326046719459⋅u ⋅Δt ⋅╲╱ 1 - u ⋅sin(t) 148205\n", | |
| "────── + ─────────────────────────────────────────────── + ───────────────────\n", | |
| "_____⎞ ⎛ ________ ________⎞ 4 \n", | |
| " 2 ⎟ ⎜ 2 ╱ 2 ╱ 2 ⎟ 17384622796800⋅u -\n", | |
| "- u ⎠ 34769245593600⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ \n", | |
| "\n", | |
| " \n", | |
| " 2 5 2 5 \n", | |
| "22100151⋅u ⋅Δt ⋅sin(t) 10831818276999618281093⋅u ⋅Δt ⋅sin(t) \n", | |
| "─────────────────────────────────── - ───────────────────────────────────── - \n", | |
| " 2 ⎛ 2⎞ \n", | |
| " 34769245593600⋅u + 17384622796800 8204334598036362240000⋅⎝1 - u ⎠ \n", | |
| " \n", | |
| "\n", | |
| " ________ \n", | |
| " 5 ╱ 2 5 \n", | |
| " 1886254549⋅u⋅Δt ⋅╲╱ 1 - u 826854427199⋅u⋅Δt \n", | |
| "──────────────────────────────────────────── - ────────────────── + ──────────\n", | |
| " ⎛ ________ ________⎞ 386324951040 \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ \n", | |
| "13414060800⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 1341406080\n", | |
| "\n", | |
| " \n", | |
| " 5 5 \n", | |
| "1886254549⋅u⋅Δt ⋅cos(t) 234341226128040406829⋅u⋅Δt 75654526\n", | |
| "────────────────────────────────── - ────────────────────────────── + ────────\n", | |
| " ⎛ ________ ________⎞ ⎛ 2⎞ \n", | |
| " ⎜ 2 ╱ 2 ╱ 2 ⎟ 482607917531550720000⋅⎝1 - u ⎠ \n", | |
| "0⋅⎝- u ⋅╲╱ 1 - u + ╲╱ 1 - u ⎠ 20510836\n", | |
| "\n", | |
| " \n", | |
| " 5 5 5 \n", | |
| "4265589533617⋅u⋅Δt ⋅cos(t) 4162893696667⋅Δt ⋅sin(t) 3844175612059⋅Δt ⋅sin(\n", | |
| "────────────────────────── + ──────────────────────── + ──────────────────────\n", | |
| " ________ 2317949706240 ⎛ 2\n", | |
| " ╱ 2 6953849118720⋅⎝1 - u \n", | |
| "49509090560000⋅╲╱ 1 - u \n", | |
| "\n", | |
| " ⎞\n", | |
| " ⎟\n", | |
| "t)⎟\n", | |
| "──⎟\n", | |
| "⎞ ⎟\n", | |
| "⎠ ⎟\n", | |
| " ⎠" | |
| ] | |
| }, | |
| "execution_count": 38, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "expr2 = sp.collect(sp.expand(expr),λ)" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 46, | |
| "metadata": {}, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/latex": [ | |
| "$\\begin{equation*}λ^{5} \\left(\\frac{u Δt^{5}}{80} - \\frac{Δt^{5} \\sin{\\left(t \\right)}}{80}\\right)\\end{equation*}$\n" | |
| ], | |
| "text/plain": [ | |
| " ⎛ 5 5 ⎞\n", | |
| " 5 ⎜u⋅Δt Δt ⋅sin(t)⎟\n", | |
| "λ ⋅⎜───── - ──────────⎟\n", | |
| " ⎝ 80 80 ⎠" | |
| ] | |
| }, | |
| "execution_count": 46, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| } | |
| ], | |
| "source": [ | |
| "expr2.args[5]" | |
| ] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Julia 1.6.3", | |
| "language": "julia", | |
| "name": "julia-1.6" | |
| }, | |
| "language_info": { | |
| "file_extension": ".jl", | |
| "mimetype": "application/julia", | |
| "name": "julia", | |
| "version": "1.6.3" | |
| }, | |
| "toc": { | |
| "base_numbering": 1, | |
| "nav_menu": {}, | |
| "number_sections": true, | |
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| "skip_h1_title": false, | |
| "title_cell": "Table of Contents", | |
| "title_sidebar": "Contents", | |
| "toc_cell": false, | |
| "toc_position": {}, | |
| "toc_section_display": true, | |
| "toc_window_display": false | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 2 | |
| } |
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