Putnam 1971 B5

B-5. Show that the graphs in the $x-y$ plane of all solutions of the system of differential equations $$x^{\prime \prime}+y^{\prime}+6 x=0, y^{\prime \prime}-x^{\prime}+6 y=0 \left({ }^{\prime}=d / d t\right)$$ which satisfy $x^{\prime}(0)=y^{\prime}(0)=0$ are hypocycloids, and find the radius of the fixed circle and the two possible values of the radius of the rolling circle for each such solution. (A hypocycloid is the path described by a fixed point on the circumference of a circle which rolls on the inside of a given fixed circle.)