Second-Order Linear Equations · Superchallenge problem

Let $y$ solve $y'' + 2y' + (1 + \pi^2)y = 0$ with $y(0) = 0$ and $y'(0) = \pi$. Let $v_k$ be the speed $|y'|$ of the mass at its $k$-th return to equilibrium ($k = 1, 2, 3, \ldots$). Compute $\displaystyle\sum_{k=1}^{\infty} v_k$.