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