= Solution
The <characteristic equation> of the reduced <homogeneous linear differential equation> is $m^2-\beta=0$. Its <polynomial roots> change type with the sign of $\beta$, giving
$$
\boxed{y(t)=
\begin{cases}
C_1e^{\sqrt\beta\,t}+C_2e^{-\sqrt\beta\,t},&\beta>0,\\[3pt]
C_1+C_2t,&\beta=0,\\[3pt]
C_1\cos(\sqrt{-\beta}\,t)+C_2\sin(\sqrt{-\beta}\,t),&\beta<0.
\end{cases}}
$$
For $\beta=0$, the zero <polynomial root> is repeated, so the second independent solution is $t$. For $\beta<0$, the <polynomial roots> are the <complex conjugate> pair $\pm i\sqrt{-\beta}$, and the displayed <sine> and <cosine> give a real <basis> of solutions. The constants are arbitrary real numbers. In every case $x(t)=y'(t)-\cos t$ reconstructs the corresponding <general solution> of the coupled system.
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