Second-Order Linear Equations · Superchallenge problem

For $b > 0$ let $y$ solve $y'' + by' + y = 0$ with $y(0) = 0$ and $y'(0) = 1$. Find the value of $b$ for which the largest value of $y(t)$ over $t \ge 0$ is exactly $e^{-\pi/4}$.