Solution
= Solution
Differentiate the second <ordinary differential equation> and use the first to eliminate $x'$:
$$
y''=x'-\sin t=\beta y+\sin t-\sin t.
$$
Thus the reduced <second-order linear differential equation> is
$$
\boxed{y''-\beta y=0}.
$$
No solutions are lost: given a solution $y$, define $x=y'-\cos t$; then both original <ordinary differential equations> hold.