Example 1. Determine the type of the equation

$$
u_{x x}-4 u_{x y}-21 u_{y y}+2 u_{x}-3 u_{y}+5 u=x^{2}
$$

and reduce it to the canonical form.
Solution.

1. Determine the coefficients of the equation $a_{11}, a_{12}$ and $a_{22}$. We have

$$
a_{11}=1, \quad a_{12}=-2, \quad a_{22}=-21
$$

2. Calculate the expression

$$
a_{12}^{2}-a_{11} a_{22}=4+21=25>0
$$

3. Since $a_{12}^{2}-a_{11} a_{22}>0$ for all $x, y$, equation (4) is of hyperbolic type in the entire $X O Y$ plane.
4. Find the general integrals of the characteristic equation:

$$
(d y)^{2}+4 d y d x-21(d x)^{2}=0
$$

Solving this equation for $d y$, we get

$$
d y=3 d x \quad \text { and } \quad d y=-7 d x
$$

Therefore, the characteristic equation (5) has two integrals:

$$
h_{1}(x, y)=y-3 x \quad \text { and } \quad h_{2}(x, y)=y+7 x .
$$

5. In equation (4), make the change of variables:

$$
x, y \longmapsto \xi=h_{1}(x, y)=y-3 x, \quad \eta=h_{2}(x, y)=y+7 x
$$

By the chain rule for differentiation,

$$
\begin{aligned}
u_{x} & \longmapsto 3 u_{\xi}+7 u_{\eta}, \quad u_{y} \longmapsto u_{\xi}+u_{\eta}, \\
u_{x x} & \longmapsto u_{\xi \xi}-54 u_{\xi \eta}+49 u_{\eta \eta}, \\
u_{x y} & \longmapsto 3 u_{\xi \xi}-54 u_{\xi \eta}+49 u_{\eta \eta}, \\
u_{y y} & \longmapsto u_{\xi \xi}+2 u_{\xi \eta}+u_{\eta \eta} .
\end{aligned}
$$

As a result of the change of variables, equation (4) takes the form $9 u_{\xi \xi}-54 u_{\eta \eta}+49 u_{\eta \eta}+12 u_{\xi \xi}-16 u_{\xi \eta}-28 u_{\eta \eta}-21 u_{\xi \xi}-$

$$
-42 u_{\xi \eta}-21 u_{\eta \eta}-6 u_{\xi}+14 u_{\eta}-3 u_{\xi}-3 u_{\eta}+5 u=\left(\frac{\eta-\xi}{10}\right)^{2}
$$

i.e., equation (4) is reduced to the canonical form

$$
u_{\xi \eta}=-\frac{10}{112} u_{\xi}+\frac{11}{112} u_{\eta}+\frac{5}{112} u-\frac{1}{112}\left(\frac{\eta-\xi}{10}\right)^{2}
$$

Answer. Equation (4) is of hyperbolic type in the entire $X O Y$ plane and has the canonical form

$$
u_{\xi \eta}=-\frac{10}{112} u_{\xi}+\frac{11}{112} u_{\eta}+\frac{5}{112} u-\frac{1}{112}\left(\frac{\eta-\xi}{10}\right)^{2}
$$

where $\xi=y-3 x$ and $\eta=y+7 x$.