Example 1. Integrate the equation

$$
2 x y d x+x^{2} d y=0
$$
Solution. The left-hand side of the given equation is the total differential of the function $u(x, y)=x^{2} y$, i.e.,

$$
2 x y d x+x^{2} d y=d\left(x^{2} y\right)
$$

The original equation can be rewritten as

$$
d\left(x^{2} y\right)=0
$$

from which we obtain its general integral

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