Example 4. Solve the equation $2 x y \ln y d x+\left(x^{2}+y^{2} \sqrt{y^{2}+1}\right) d y=0$.
Solution. Here $M=2 x y \ln y, N=x^{2}+y^{2} \sqrt{y^{2}+1}$. We have

$$
\frac{(\partial N / \partial x)-(\partial M / \partial y)}{M}=\frac{2 x-2 x(\ln y+1)}{2 x y \ln y}=-\frac{1}{y},
$$

therefore,

$$
\frac{d \ln \mu}{d y}=-\frac{1}{y}, \quad \mu=\frac{1}{y}
$$

The equation

$$
\frac{2 x y \ln y d x}{y}+\frac{x^{2}+y^{2} \sqrt{y^{2}+1}}{y} d y=0
$$

is an equation in total differentials. It can be written in the form $d\left(x^{2} \ln y\right)+y \sqrt{y^{2}+1} d y=0$, from which

$$
x^{2} \ln y+\frac{1}{3}\left(y^{2}+1\right)^{3 / 2}=C
$$