Example 4.16. Find the general solution of the equation

$$
y^{\prime \prime}-5 y^{\prime}-6 y=0
$$
Solution. Let's write down the characteristic equation; for this, we replace the function $y$ and its derivatives $y^{\prime}$ and $y^{\prime \prime}$ with the corresponding powers of $r: r^{0}=1, r u r^{2}$. Then we get

$$
r^{2}-5 r-6=0
$$

from which $r_{1}=-1, r_{2}=6$. Since the roots of the characteristic equation are real and distinct, the general solution of the given differential equation, according to formula (3), is:

$$
y=C_{1} e^{-x}+C_{2} e^{6 x}
$$