Chapter 7.5, Problem 8P

(a) Show that the system $\frac{dx}{dt}=y+\frac{xf\left(r\right)}{r},\frac{dy}{dt}=-x+\frac{yf\left(r\right)}{r}$ has periodic solutions corresponding to the zeros of $f\left(r\right)$. What is the direction of motion on the closed trajectories in the phase plane ?

(b) Let $f\left(r\right)=r{\left(r-6\right)}^2\left(r^2-6r+5\right)$. Determine all periodic solutions and determine their stability characteristics.