Numerical solution to differential equation I want to find a numerical solution to this differential equation:
$$x''-\frac{2kqQ}{m\left(x^2+(\frac{ab}{2})^2\right)}=0$$
$k,q,Q,a,b,m$ are constants.
Can anyone recommend a program or a method for solving this? I have mathematica but I have no idea how to solve this in mathematica.
 A: Mathematica example:
Manipulate[Module[{sol},
  sol=First@NDSolve[{
    x''[t]-A/(x[t]^2+c^2)==0,
    x[0]==1,x'[0]==0
  },x[t],{t,0,10}];
  Plot[x[t]/.sol,{t,0,10}]],
{{A,1,"(2kqQ)/m"},0,10},
{{c,1,"ab/2"},0,10}]

Also, your equation is pretty solvable by hand.
Let $A = \frac{2kqQ}{m}, c = \frac{ab}{2}$. If you multiply your system with $x'$ and integrate that, you'll get the first integral of the system:
$$
x''x' - \frac{Ax'}{x^2+c^2} = 0\\
\frac{(x')^2}{2} - \int \frac{A dx}{x^2+c^2} = E\\
\frac{(x')^2}{2} - \frac{A}{c}\arctan \frac{x}{c} = E\\
\int\frac{dx}{\sqrt{2E + \frac{2A}{c}\arctan \frac{x}{c}}} = t - t_0\\
$$
A: Just another way (leading to the same result as @uranix.
To make notations simpler, let
$$\alpha=\frac{2kqQ}{m}\qquad \text{and} \qquad \beta=\frac{a b}{2}$$ making the equation to be
$$x''-\frac{\alpha}{x^2+\beta^2}=0$$ Switching variables
$$\frac{t''}{[t']^3}+\frac{\alpha}{x^2+\beta^2}=0$$ Reduction of order $p=t'$ gives
$$p=\pm \frac{\sqrt{\beta }}{\sqrt{2 \alpha  \tan ^{-1}\left(\frac{x}{\beta }\right)+ c_1}}$$ No way to go further even if $c_1=0$.
