Example 2. Find the general solution of the equation

$$
y^{\prime \prime \prime}+2 y^{\prime \prime}+y^{\prime}=0
$$
Solution. The characteristic equation is

$$
\lambda^{3}+2 \lambda^{2}+\lambda=0 .
$$

From this, $\lambda_{1}=\lambda_{2}=-1, \lambda_{3}=0$. The roots are real, and one of them, namely $\lambda=-1$, is a double root, so the general solution is

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