Example 5. Solve the system by D'Alembert's method

$$
\left\{\begin{array}{l}
\frac{d x}{d t}=5 x+4 y+e^{t} \\
\frac{d y}{d t}=4 x+5 y+1
\end{array}\right.
$$
Solution. We choose $\lambda$ according to formula (19): $4+5 \lambda=\lambda(5+4 \lambda)$, from which $\lambda_{1,2}= \pm 1$. Then, by formula (20) for the case $\lambda=1$, we have

$$
\begin{aligned}
& x+y=e^{(s+4 \cdot 1) t}\left\{C_{1}+\int\left(e^{t}+1\right) e^{-(s+4 \cdot 1) t} d t\right\}= \\
& \quad=e^{9 t}\left\{C_{1}+\int\left(e^{-8 t}+e^{-9 t}\right) d t\right\}=C_{1} e^{9 t}-\frac{1}{8} e^{t}-\frac{1}{9}
\end{aligned}
$$

For $\lambda=-1$, we similarly obtain

$$
x-y=e^{(5-4) t}\left\{C_{2}+\int\left(e^{t}-1\right) e^{-(5-4) t} d t\right\}=C_{2} e^{t}+t e^{t}+1
$$

Thus, we have two first independent integrals of system (21):

$$
\left(x+y+\frac{1}{8} e^{t}+\frac{1}{9}\right) e^{-9 t}=C_{1}, \quad\left(x-y-t e^{t}-1\right) e^{-t}=C_{2}
$$

The integration of the system is complete.

Remark. If the right-hand sides of the normal system of equations have the form $\frac{a x+b y+c z+P(t)}{t}$, where $a, b, c$ are constants and $P(t)$ is a polynomial in $t$, then the substitution $t=e^{\tau}$ reduces the system to one with constant coefficients.