= Solution
The original PDF has the absolute-value hypothesis $|g(\xi)|<Ce^{-s\xi}$, which is missing from the local OCR. From the <Green's function> formula and $|\sin|\leq1$,
$$
\begin{aligned}
|y(x)|&\leq\frac{C}{b}\int_0^xe^{-a(x-\xi)}e^{-s\xi}\,d\xi\\
&=\frac{C}{b}e^{-ax}\int_0^xe^{-(s-a)\xi}\,d\xi\\
&=\frac{C}{b(s-a)}e^{-ax}[1-e^{-(s-a)x}]\leq\frac{C}{b(s-a)}e^{-ax}.
\end{aligned}
$$
Because $s>a>0$, the upper bound tends to zero. The <squeeze theorem> proves $\boxed{y(x)\to0\text{ as }x\to\infty}$.
Back to article page
