Example 2. Find the general solution of the linear equation $x y^{\prime}-y=x^{3}$.
Solution. Let $y=u v$, then $y^{\prime}=u^{\prime} v+u v^{\prime}$. Rewrite the equation as

$$
x\left(u^{\prime} v+u v^{\prime}\right)-u v=x^{3} \text { or } u\left(x v^{\prime}-v\right)+x u^{\prime} v=x^{3}
$$

Choose $v$ such that the expression in parentheses becomes zero, then

$$
x \frac{d v}{d x}-v=0, \frac{d v}{v}=\frac{d x}{x}, \ln v=\ln x, v=x
$$

Equation (A) with $v=x$ can be written as:

$$
x^{2} d u=x^{3} d x, d u=x d x
$$

from which

$$
u=\frac{x^{2}}{2}+C
$$

Therefore,

$$
y=u v=\left(\frac{x^{2}}{2}+C\right) x
$$

is the general solution.