# I wanted to know how I could solve this differential equation: $d^2x/dt^2 = -k(x^2+p^2)/(x^2-p^2)^2$

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.

• – Angina Seng Oct 29 '18 at 4:02

## 1 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.