Second-Order Linear Equations · Superchallenge problem

The oscillator $y'' + by' + \left(1 + \dfrac{b^2}{4}\right)y = 0$, $b > 0$, has energy $E = \tfrac12 y'^2 + \tfrac12\left(1 + \tfrac{b^2}{4}\right)y^2$. Between any two consecutive local maxima of $y$ it loses exactly three quarters of its energy. Find $b$.