Here is an unusual way to solve the two-dimensional isotropic oscillator -- the motion of a particle subject to a force -.
Show that by choosing a suitable rotating reference frame, you can arrange that the centrifugal force exactly cancels the force .
Recalling the analogy between the Coriolis and magnetic forces, you should be able to write down the general solution for the motion as seen in the rotating frame.
If you write your solution in the complex form of Section 2.7, then you can transform back to the nonrotating frame by multiplying by a suitable rotating complex number.
Show that the general solution is an ellipse. [See Problem 8.11 for some guidance on this last part.]