Example 1. Find the general solution of the equation

$$
y^{\prime \prime \prime}-2 y^{\prime \prime}-3 y^{\prime}=0
$$
Solution. We form the characteristic equation

$$
\lambda^{3}-2 \lambda^{2}-3 \lambda=0
$$

We find its roots: $\lambda_{1}=0, \lambda_{2}=-1, \lambda_{3}=3$. Since they are real and distinct, the general solution is

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