Example 1. Reduce the system of equations to canonical form

$$
\left\{\begin{aligned}
y_{2} y_{1}^{\prime}-\ln \left(y_{1}^{\prime \prime}-y_{1}\right) & =0 \\
e^{y_{2}^{\prime}}-y_{1}-y_{2} & =0
\end{aligned}\right.
$$
Solution. The given system is of the third order, since $k_{1}=2, k_{2}=1$, and thus $p=3$. Solving the first equation for $y_{1}^{\prime \prime}$ and the second for $y_{2}^{\prime}$, we obtain the canonical system

$$
y_{1}^{\prime \prime}=y_{1}+e^{y_{2} y_{1}^{\prime}}, \quad y_{2}^{\prime}=\ln \left(y_{1}+y_{2}\right)
$$

A system of ordinary differential equations of the first order of the form

$$
\frac{d x_{k}}{d t}=f_{k}\left(t, x_{1}, x_{2}, \ldots, x_{n}\right), \quad k=1,2, \ldots, n
$$

where $t$ is the independent variable; $x_{1}, x_{2}, \ldots, x_{n}$ are unknown functions of $t$, is called a normal system.

The number $n$ is called the order of the normal system (3). Two systems of differential equations are called equivalent if they have the same solutions.

Any canonical system (2) can be reduced to an equivalent normal system (3), and the order of these systems will be the same.