Example 1. Solve the equation $y\left(y^{\prime}\right)^{2}+(x-y) y^{\prime}-x=0$.
Solution. We will solve this equation with respect to $y^{\prime}$:

$$
y^{\prime}=\frac{y-x \pm \sqrt{(x-y)^{2}+4 x y}}{2 y} ; \quad y^{\prime}=1, \quad y^{\prime}=-\frac{x}{y},
$$

from which

$$
y=x+C, \quad y^{2}+x^{2}=C^{2}
$$