a) Proof:
Consider the equations ## \dot{a}=-\epsilon\sin\theta h(a\cos\theta, a\sin\theta) ## and ## \dot{\theta}=-1-\frac{\epsilon}{a}\cos\theta h(a\cos\theta, a\sin\theta) ##.
Let ## \theta(t)=\psi(t)-t ##.
Then ## \dot{\theta}(t)=\dot{\psi}(t)-1 ##.
By direct substitution of ##...