Example 4.18. Find the general solution of the equation

$$
y^{\prime \prime}+9 y=0
$$
Solution. This equation corresponds to the characteristic equation

$$
r^{2}+9=0
$$

which has two imaginary conjugate roots $r_{1,2}= \pm 3 i$. Using formula (5) with $\alpha=0$ and $\beta=3$, we obtain the general solution

$$
y=C_{1} \cos 3 x+C_{2} \sin 3 x
$$