Second-Order Linear Equations · Challenge problem

A $1$ kg mass on a spring with constant $20$ N/m and a dashpot with damping constant $4$ N$\cdot$s/m obeys $y'' + 4y' + 20y = 0$. It is struck at equilibrium, $y(0) = 0$, $y'(0) = 8$ m/s. Its energy is $E = \tfrac12 y'^2 + 10y^2$ joules. How much energy does the dashpot absorb during the first full oscillation, that is, until the mass next passes through equilibrium moving in the positive direction?