Solution
= Solution
A <central force> conserves <angular momentum>
$$
\ell=mr^2\dot\theta.
$$
Put $u=1/r$. Since $\dot\theta=\ell u^2/m$,
$$
\dot r=-\frac{\ell}{m}u',
\qquad
\ddot r=-\frac{\ell^2}{m^2}u^2u'',
\qquad
r\dot\theta^2=\frac{\ell^2}{m^2}u^3,
$$
where primes now denote <differentiation> with respect to $\theta$. Substitution into the radial equation gives the <Binet equation>
$$
\boxed{
u''+u
=\frac{m}{\ell^2u^2}
\frac{dV}{dr}\bigg|_{r=1/u}}.
$$