Second-Order Linear Equations · Superchallenge problem

For $b > 0$ let $y$ solve $y'' + by' + y = 0$ with $y(0) = 1$, $y'(0) = 0$, and let $J(b) = \displaystyle\int_0^\infty \big(y^2 + y'^2\big)\,dt$. Find the minimum value of $J(b)$ over all $b > 0$.