Variation of constants

Homogeneous equation. $y' - y/x = 0$ gives
$$\frac{y'}{y} = \frac1x \implies \ln|y| = \ln x + \text{const} \implies y_h(x) = Cx.$$

Variation of constants. Set $y(x) = C(x)x$. Then $y' = C'x + C$, and
$$y' - \frac{y}{x} = C'x + C - C = C'x.$$
We need $C'x = x^2$, so $C'(x) = x$, hence $C(x) = \frac{x^2}{2} + K$.

General solution.
$$y(x) = \left(\frac{x^2}{2} + K\right)x = \frac{x^3}{2} + Kx, \qquad K \in \mathbb{R}.$$

Remark. This is the standard method of variation of constants.