Example 1. Solve the equation $y^{\prime}+x y=y^{3} e^{x^{2}}$.
Solution. Let $y=u \cdot v, y^{\prime}=u^{\prime} v+u v^{\prime}$. We get $u^{\prime} v + u v^{\prime} + x u \cdot v = u^{3} \cdot v^{3} \cdot e^{x^{2}}$.

1) From the equation $u v^{\prime} + x u v = 0$, i.e., $\frac{d v}{d x} = -x v$, or $\frac{d v}{v} = -x d x$, we find $\ln |v| = -\frac{x^{2}}{2}$, or $v = e^{-x^{2} / 2}$ (the constant of integration is taken to be zero).
2) The remaining equation $u^{\prime} v = u^{3} v^{3} e^{x^{2}}$ after substituting $v = e^{-x^{2} / 2}$ becomes $u^{\prime} = u^{3}$, or $-\frac{d u}{u^{3}} = -d x$. From this, $\frac{1}{2 u^{2}} = -x + C$, or $u = \frac{1}{\sqrt{2(C - x)}}$.

From the obtained solutions $u$ and $v$, we form the general solution of the given equation.

Answer. $y = \frac{e^{-x^{2} / 2}}{\sqrt{2(C - x)}}$.