Example 4.19. Find the general solution of the equation

$$
y^{\prime \prime}+6 y^{\prime}+25 y=0 \text {. }
$$
Solution. The characteristic equation

$$
r^{2}+6 r+25=0
$$

has two complex conjugate roots $r_{1,2}=-3 \pm 4 i$. Using formula (5) with $\alpha=-3$ and $\beta=4$, we obtain the general solution

$$
y=e^{-3 x}\left(C_{1} \cos 4 x+C_{2} \sin 4 x\right)
$$