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# markusbuchholz/latex_eq_2.md

Created Jul 12, 2021

eq1--------------------------------------------

$$\large \frac{d^2}{dt^2}I(t) + \frac{R}{L}\frac{d}{dt}I(t)+ \frac{1}{LC}I(t)=0$$

eq2--------------------------------------------

$$\large F\Bigg(t, x(t),\frac{dx(t)}{dt}, \frac{d^2x(t)}{dt^2}, \frac{d^nx(t)}{dt^n} \Bigg)=0$$

eq3--------------------------------------------

$$\large \frac{d^nx(t)}{dt^n} = f\Bigg(t, x(t),\frac{dx(t)}{dt}, \frac{d^2x(t)}{dt^2}, \frac{d^{n-1}x(t)}{dt^{n-1}} \Bigg)$$

eq4--------------------------------------------

$$\large \begin{cases} x(t_{0}) = x_{0}\ \ \frac{dx(t)}{dt} = f\Big(t,x(t)\Big) \end{cases}$$

eq5--------------------------------------------

$$\large \frac{dx(t)}{dt} = \lim_{h \to 0 } \frac{x(t+h)-x(t)}{h} \approx \frac{x(t+h)-x(t)}{h}$$

eq6--------------------------------------------

$$\large x(t +\Delta t) = x(t) + \Delta t f\Big(t, x(t)\Big)$$

eq7--------------------------------------------

$$\large \frac{x(t)}{dt}\Big(t+0.5\Delta \Big) \approx \frac{x(t + 0.5\Delta t + 0.5\Delta t)-x(t + 0.5\Delta t - 0.5\Delta t)}{\Delta t}$$ $$\large= \frac{x(t + \Delta t)-x(t)}{\Delta t}$$

eq8--------------------------------------------

$$\large x(t +\Delta t) \approx x(t) + \Delta t f\Bigg(t + 0.5\Delta t, x\Big(t + 0.5\Delta t\Big)\Bigg)$$

eq9--------------------------------------------

$$\large \frac{dx(t)}{dt} \approx \frac{x(t + 0.5\Delta t) - x(t)}{0.5\Delta t}$$

eq10--------------------------------------------

$$\large x(t +\Delta t) \approx x(t) + \Delta t f\Bigg(t + 0.5\Delta t, x(t) + f\Big(t + x(t)\Big)\Bigg)$$

eq11--------------------------------------------

$$\large k_{1} = \Delta t f\Big(t, x(t)\Big),$$

$$\large k_{2} = \Delta t f\Big(t + 0.5\Delta t, x(t) + 0.5k_{1}\Big),$$

$$\large k_{3} = \Delta t f\Big(t + 0.5\Delta t, x(t) + 0.5k_{2}\Big),$$

$$\large k_{4} = \Delta t f\Big(t + \Delta t, x(t) + k_{3}\Big),$$

$$\large x(t +\Delta t) = x(t) + \frac{1}{6}\Big(k_{1} + 2k_{2} + 2k_{3} + k_{4} \Big)$$

eq12--------------------------------------------

$$\large \frac{dx(t)}{dt} = x(t) + t$$

eq13--------------------------------------------

$$\large x(t) = Ce^t - t - 1$$

eq14--------------------------------------------

$$\large \frac{d^2x(t)}{dt^2} = a\Big(t, x(t)\Big)$$

eq15--------------------------------------------

$$\large x(t +\Delta t) = 2x(t) - x(t -\Delta t) + a\Big(t, x(t)\Big)\Delta t^2$$

eq16--------------------------------------------

$$\large x(t +\Delta t) \approx x(t) + v(t)\Delta t + 0.5a\Big(t, x(t)\Big)\Delta t^2$$

eq17--------------------------------------------

$$\large v(t +\Delta t) \approx v(t) + 0.5\Big(a(t, x(t)\Big) + a\Big(t + \Delta t, x(t + \Delta t) \Big)\Delta t$$

eq18--------------------------------------------

$$\large \frac{d^2x(t)}{dt^2} + \omega_{0}^2x = 0$$

eq19--------------------------------------------

$$\large x(t) = Acos(\omega_{0}t + \phi)$$

eq20--------------------------------------------

$$\large x(t) = - A\omega_{0}sin(\omega_{0}t + \phi)$$

eq21--------------------------------------------

$$\large f(x) = 0$$

eq22--------------------------------------------

$$\large ax^3 + bx^2 = cx + d$$

eq23--------------------------------------------

$$\large f(x) = f(x_{0}) + (x - x_{0})f'(x_{0}) + \frac{(x - x_{0})^2}{2!}f''(x_{0}) + ...$$

eq24--------------------------------------------

$$\large x_{i} - x_{i-1} = - \frac{f(x_{i-1})}{f'(x_{i-1})}$$

eq25--------------------------------------------

$$\large \frac {1}{e^{-x} - 0.5} = 0$$ $$\large \frac {1}{1 + e^{-x}} - 0.5 = 0$$

eq26--------------------------------------------

$$\large 2sin(x) - cos(x)^2 = 0$$

eq27--------------------------------------------

$$\large x^2 - 9 = 0$$