Example 1. Integrate the system of equations

$$
\left\{\begin{array}{l}
\frac{d x}{d t}=y+1 \\
\frac{d y}{d t}=x+1
\end{array}\right.
$$
Solution. From the first equation of the system (3), we find $y=\frac{d x}{d t}-1$, then

$$
\frac{d y}{d t}=\frac{d^{2} x}{d t^{2}}
$$

Substituting (4) into the second equation of the system (3), we obtain a second-order linear differential equation with constant coefficients

$$
\frac{d^{2} x}{d t^{2}}-x-1=0
$$

The general solution of equation (5)

$$
x=C_{1} e^{t}+C_{2} e^{-t}-1
$$

Finding the derivative with respect to $t$ from (6), we get

$$
y=\frac{d x}{d t}-1=C_{1} e^{t}-C_{2} e^{-t}-1
$$

The general solution of the system (3):

$$
x=C_{1} e^{t}+C_{2} e^{-t}-1, \quad y=C_{1} e^{t}-C_{2} e^{-t}-1 .
$$