Derive the equation

$$\frac{d^{2} u}{d \theta^{2}}+u=\frac{f(u)}{m h^{2} u^{2}}$$

for the motion of a particle of mass $m$ under an attractive central force $f$, where $u=1 / r$ and $r$ is the distance of the particle from the centre of force, and where $m h$ is the angular momentum of the particle about the centre of force.

[Hint: you may assume the expressions for the radial and transverse accelerations in the form $\ddot{r}-r \dot{\theta}^{2}, 2 \dot{r} \dot{\theta}+r \ddot{\theta}$.]