Consider the motion of a particle of mass m in a 3D isotropic harmonic oscillator potential $\displaystyle V(r)=\frac{1}{2}kr^2$ where k > 0

Determine the parametric equations for the orbit of the particle using the general differential equations of motions

$\displaystyle f(r)=m(\ddot{r}-\frac{L^2}{m^2r^3})$

and $\displaystyle \dot{\theta}=\frac{L}{mr^2}$

Assume that $\displaystyle E>E_{0}\equiv{E_{min}},r(0)=r_{max},\theta(0)=0$ and L is not zero.

The final answer should not contain $\displaystyle r_{max}$; the final result should be given in terms of the parameters m, k, E and L.