What is the equation for a body's trajectory in a central force field?

[neworder1]

Homework Statement

Show that in a central force field $$\vec{F}(\vec{r})=f(r)\frac{\vec{r}}{r}$$ the equation for a body's trajectory in terms of $$u=\frac{1}{r}$$ is given by:
$$\ddot{u}+u=-\frac{mf(\frac{1}{u})}{L^{2}u^{2}}$$

where m is the body's mass and L is the angular momentum

Homework Equations

The Attempt at a Solution

Transforming Newton's equations for $$\ddot{r}$$ leads to ab equation for $$\ddot{u}$$ and $$u$$ where unfortunately $$\dot{u}$$ also appears and I don't know how to eliminate it from the equation.

 

[malawi_glenn]

can you show us how you did it?

 

[azatkgz]

$$\ddot u=\frac{d^2u}{d\theta^2}$$