A classical mechanics problem by Rohith M.Athreya

The differential equation for central motion $\ddot{\mathbf{r}} \; = \; -\frac{K}{r^n}\hat{\mathbf{r}}$ becomes $\ddot{r} - r\dot{\theta}^2 \; = \; -Kr^{-n} \hspace{2cm} r^2\dot{\theta} = h$ for some constant $h$ , after introducing polar coordinates. Thus $\ddot{r} - h^2r^{-3} \; = \; -Kr^{-n}$ If we put $r = u^{-1}$ then $\dot{r} = -h\frac{du}{d\theta} \hspace{2cm} \ddot{r} = -h^2u^2\frac{d^2u}{d\theta^2}$ so we obtain the differential equation $\frac{d^2u}{d\theta^2} + u \; = \; \tfrac{K}{h^2}u^{n-2}$ The solution $r = r_0\cos\theta$ , so $u = r_0^{-1}\sec\theta$ , is possible provided that $r_0^{-1}\big(\sec\theta\tan^2\theta + \sec^3\theta\big) + r_0^{-1}\sec\theta \; = \; \frac{K}{h^2r_0^{n-2}} \sec^{n-2}\theta$ which is possible provided that $n = \boxed{5}$ and $K = 2h^2r_0^2$ .