A variant of the ode $\dot x=y, \dot y=-x$

Note that $$ \frac{d}{dt}{(r^2)}=\frac{d}{dt}{(x^2+y^2)}=2x\dot{x}+2y\dot{y}=f(r^2)\left(2xy-2yx\right)=0, $$ so each solution still lies at a constant radius... the only difference between this and the original equation is that the frequency of the motion now varies with $r$. You can easily compute $\omega(r)$ in terms of $f$.

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Q:Related rates problem