I am quite stuck on how do I solve this differential equation, I am not an expert in differential equations, I formed it for solving a physics problem, I have never really encountered such equations and am quite stuck on where to start solving it from or how do I simplify it to variable separable, I would appreciate that if anyone who knows the solution, to explain it step-wise (for as I mentioned I am not an expert in solving such equations): the equation is:
$$\frac{\mathrm{d}^2x}{\mathrm{dt}^2} = -k \frac{x^2+p^2}{(x^2-p^2)^2}$$
Apologies if this question has already been asked, for I searched it but couldn't find it, I am also sorry but I am not quite sure how to use MathJax, so I have so I have written it this way.
Best Answer
The following method is useful when the independent variable ($t$ in this case) does not appear explicitly in the equation. We let $y=dx/dt$ and consider $y$ as a function of $x$. $$ \frac{d^2x}{dt^2}=\frac{dy}{dt}=\frac{dy}{dx}\,\frac{dx}{dt}=y\,\frac{dy}{dx}. $$ The equation becomes $$ y\,\frac{dy}{dx}= -k\,\frac{x^2+p^2}{(x^2-p^2)^2}. $$ Integrating we get $$ \frac12\,y^2=-k\int\frac{x^2+p^2}{(x^2-p^2)^2}\,dx. $$ From here you can get $y$ as a function of $x$, say $y=F(x,C)$ (where $C$ is a constant of integration.) Then you get the solution of the original problem by solving the first order equation $$ y=\frac{dx}{dt}=F(x,C). $$ Computations are usually quite complicated.