= Solution
Use the zero initial displacement and velocity given in the PDF. The characteristic roots of the homogeneous equation are $-1\pm i\mu$. Its <causal Green function of a damped oscillator> is
$$
G(t)=H(t)\frac{e^{-t}\sin(\mu t)}{\mu},
$$
where $H$ is the <Heaviside step function>. The positive-time kernel has $G(0+)=0$ and $G'(0+)=1$, so its <derivative> jump generates a unit impulse. The zero-data forced solution is consequently the causal <convolution>
$$
\boxed{x(t)=\frac1\mu\int_0^te^{-(t-s)}\sin[\mu(t-s)]g(s)\,ds,\qquad t\geq0.}
$$
Differentiating once gives $x'(0)=0$; differentiating twice supplies the endpoint term $g(t)$ and verifies the differential equation. No homogeneous transient is required with these initial data.
Back to article page
