Second-Order Linear Equations · Superchallenge problem

A mass obeys $y'' + by' + y = 0$ with damping constant $b > 0$. It starts at $y(0) = 1$ moving toward equilibrium with velocity $y'(0) = -3$. For small $b$ it certainly passes through equilibrium ($y = 0$ at some $t > 0$); for large $b$ it creeps back without crossing. Find the supremum of the values of $b$ for which the mass passes through equilibrium.