Central Force Motion: Solve r(t) & theta(t)

[roeb]

Homework Statement

Consider the motion of a particle in the central force F(r) = -kr.
Solve for the particle's location as a function of time, r(t) and theta(t).

Homework Equations

The Attempt at a Solution

$$E = 1/2 m r'^2 + \frac{L^2}{2mr^2} + U(r)$$
I know U(r) = 1/2 k r^2
$$\frac{dr}{dt} = \sqrt{\frac{2}{m}(E-U(r)) - \frac{L^2}{m^2 r^2}}$$
(Where L is ang. momentum)

Plugging in U(r) I get a really nasty integral

$$dt = \frac{r^2 dr}{\sqrt{2E/m r^2 - k/m r^4 - L^2/m^2}}$$

According to my professor I can use a trig sub to solve this, but I am not getting anywhere.
Is there some sort of relationship that I am missing? I know it's supposed to be an ellipse but I seem to get any sort of substitutions to work.

 

[weejee]

This integral is such a mess.
I think the easiest way is to solve this problem in cartesion coordinates and convert it to polar coordinates.