Example 3. Solve the equation $y^{\text {IV }}-16 y=0$.
Solution. The characteristic equation

$$
r^{4}-16=0 \text { or }\left(r^{2}-4\right)\left(r^{2}+4\right)=0
$$

has roots

$$
r_{1}=2, r_{2}=-2, r_{3}=2 i, r_{4}=-2 i
$$

The imaginary conjugate roots $r_{3}$ and $r_{4}$ (for which $\alpha=0, \beta=2$) correspond to the particular solutions

$$
y_{3}=\cos 2 x, y_{4}=\sin 2 x
$$

obtained from formulas (11.13).

Thus, the general solution is

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