Second-Order Linear Equations · Superchallenge problem

Every solution of $y'' + by' + cy = 0$ tends to $0$ as $t \to \infty$ (along with its derivative). The solution with $y(0) = 0$, $y'(0) = 1$ satisfies $\displaystyle\int_0^\infty y\,dt = \dfrac15$ and $\displaystyle\int_0^\infty y^2\,dt = \dfrac{1}{20}$. Find the quasi-period of the oscillation.