Example 12. Find a particular solution of the equation

$$
y^{\prime \prime}+4 y^{\prime}+5 y=8 \cos x
$$

bounded as $x \rightarrow-\infty$.
Solution. The general solution of the given equation is

$$
y=e^{-2 x}\left(C_{1} \cos x+C_{2} \sin x\right)+2(\cos x+\sin x)
$$

As $x \rightarrow-\infty$, the quantity $e^{-2 x} \rightarrow+\infty$ and for any $C_{1}$ and $C_{2}$, not both zero, the first term on the right-hand side (26) will be an unbounded function as $x \rightarrow-\infty$, while the second term is a bounded function for all values of $x$. Therefore, only when $C_{1}=C_{2}=0$ do we have a solution of equation (25) that is bounded as $x \rightarrow-\infty$, namely

$$
y=2(\cos x+\sin x)
$$

Moreover, the solution (27) of equation (25) is bounded for all $x$:

$$
|y|=|2(\cos x+\sin x)| \leqslant 2(|\cos x|+|\sin x|)<4
$$

for all $x \in(-\infty,+\infty)$.