Example 1. Find the general solution of the equation $y^{\prime \prime}-5 y^{\prime}+6 y=0$ and identify the particular solution that satisfies the initial conditions: $y=1, y^{\prime}=2$ when $x=0$.
Solution. The characteristic equation (10.16) for the given equation takes the form $k^{2}-5 k+6=0$.

Since this equation has distinct real roots $k_{1}=2, k_{2}=3$, the general solution of the differential equation, according to formula (10.17), is:

$$
y=C_{1} e^{2 x}+C_{2} e^{3 x}
$$

To find the particular solution, we substitute the initial conditions into the expressions for $y$ and $y^{\prime}$, where

$$
y^{\prime}=2 C_{1} e^{2 x}+3 C_{2} e^{3 x}
$$

Substituting these data leads to the system of equations:

$$
\left.\begin{array}{r}
1=C_{1}+C_{2} \\
2=2 C_{1}+3 C_{2},
\end{array}\right\}
$$

from which the values of the arbitrary constants are determined: $C_{1}=1$, $C_{2}=0$. Therefore, the particular solution is given by the formula

$$
y=e^{2 x}
$$