Example 3. Find the general solution of the equation $y^{\prime \prime \prime}+4 y^{\prime \prime}+13 y^{\prime}=0$.
Solution. The characteristic equation

$$
\lambda^{3}+4 \lambda^{2}+13 \lambda=0
$$

has roots $\lambda_{1}=0, \lambda_{2}=-2-3 i, \lambda_{3}=-2+3 i$.

The general solution is

$$
y_{0.0}=C_{1}+C_{2} e^{-2 x} \cos 3 x+C_{3} e^{-2 x} \sin 3 x
$$