Second-Order Linear Equations · Superchallenge problem

Between blows, a mass obeys $y'' + 2y' + (1 + 4\pi^2)y = 0$. It starts at rest at equilibrium, and at each of the times $t = 0, \tfrac12, 1, \tfrac32, 2, \ldots$ a hammer blow instantly increases its velocity by $1$ (without changing its position). Let $v_n$ be the velocity just after the $n$-th blow. Find $\displaystyle\lim_{n\to\infty} v_n$.