Consider a particle in a Newtonian potential \(V(r)=-k / r+\epsilon / r^{n}\) for some integer \(n\). Using the alternate variable \(u=1 / r\),

(a) show that the radial equation of motion can be put into the form

\[\frac{d^{2} u}{d \varphi^{2}}+u=\frac{1}{p}+n \kappa u^{n+1}\]

where

\[p \equiv \frac{\ell^{2}}{\mu k} \quad, \quad \kappa \equiv \frac{\mu k}{\ell^{2}}\]

\(\mu\) is the reduced mass and \(\ell\) is angular momentum.

(b) We want to find \(u(\varphi)\), the shape of the trajectory. Write numerical code that solves this equation, taking as input \(\mu, k, \epsilon, n\), and \(\ell\). Study various scenarios, including (1) \(n=2\) and \(\epsilon\) large, and (2) \(n=3\) with \(\epsilon\) small.