= Solution
The initial conditions in the underdamped solution give $A=x_0$ and $B=\gamma x_0/\Omega$. Consequently
$$
x=x_0e^{-\gamma t}\left(\cos\Omega t+\frac\gamma\Omega\sin\Omega t\right),\qquad
\dot x=-\frac{x_0\omega_0^2}{\Omega}e^{-\gamma t}\sin\Omega t.
$$
The exact <mechanical energy> is therefore
$$
E(t)=E_0e^{-2\gamma t}\left[1+\frac\gamma\Omega\sin2\Omega t+\frac{2\gamma^2}{\Omega^2}\sin^2\Omega t\right].
$$
For weak damping, $\gamma\ll\omega_0$, the bracket differs from one by a bounded relative correction of order $\gamma/\omega_0$. To leading order the energy and its half-life are
$$
\boxed{E(t)\simeq E_0e^{-2\gamma t},\qquad t_{1/2}\simeq\frac{\ln2}{2\gamma}.}
$$
The exact identity $\dot E=-2m\gamma\dot x^2\le0$ confirms <dissipation>. The small rapid variation of the exponential-normalized energy does not mean that total energy increases; the half-life formula is a weak-damping approximation, not an exact equality for arbitrary $\gamma$.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017-resit-oscillator-energy-and-resonance.png]
{title=Damped oscillator displacement and energy, with undamped resonant forcing}
{description=Original dimensionless model curves. The exact damped energy is compared with its weak-damping exponential approximation; resonant growth is the ideal prediction without damping.}
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