= Solution
Set $\omega_d=\sqrt{\omega_0^2-\gamma^2/4}>0$. The real solution of the <damped harmonic oscillator> equation is $x-x_0=e^{-\gamma t/2}(A\cos\omega_dt+B\sin\omega_dt)$. The initial displacement fixes $A=a-x_0$, and release from rest gives $B=\gamma A/(2\omega_d)$. Therefore
$$
\boxed{x(t)=x_0+(a-x_0)e^{-\gamma t/2}
\left(\cos\omega_dt+\frac{\gamma}{2\omega_d}\sin\omega_dt\right).}
$$
The <exponential decay> rate of the amplitude is $\gamma/2$, while the weak-damping energy decay rate is $\gamma$.
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