Example 3. Convert the differential equation

$$
\frac{d^{2} x}{d t^{2}}+p(t) \frac{d x}{d t}+q(t) x=0
$$

to a normal system.
Solution. Let $x=x_{1}, \frac{d x}{d t}=x_{2}$, then $\frac{d x_{1}}{d t}=x_{2}, \frac{d^{2} x}{d t^{2}}=\frac{d x_{2}}{d t}$. Substituting these expressions into the given equation, we get

$$
\frac{d x_{2}}{d t}+p(t) x_{2}+q(t) x_{1}=0
$$

The normal system will have the form

$$
\frac{d x_{1}}{d t}=x_{2}, \quad \frac{d x_{2}}{d t}=-p(t) x_{2}-q(t) x_{1}
$$

A solution of the system (3) in the interval $(a, b)$ is any set of $n$ functions

$$
x_{1}=\varphi_{1}(t), \quad x_{2}=\varphi_{2}(t), \quad \ldots, \quad x_{n}=\varphi_{n}(t)
$$

defined and continuously differentiable in the interval $(a, b)$, if they turn the equations of the system (3) into identities valid for all values $t \in(a, b)$.