4

Use Reduction of Order. With $y = x u$, $y' = u + x u'$, $y'' = 2 u' + x u''$, the equation becomes $$x^3 u'' + 4 x^2 u' = x^3$$, or (with $v = u'$), $$v' + (4/x) v = 1$$. The general solution to this first-order linear differential equation is $v = x/5 + C/x^4$. Integrating, we get $$u = x^2/10 + C_1/x^3 + C_2$$ (where $C_1 = -C/3$). Thus the general solution is

$$y = \frac{x^3}{10} + \frac{C_1}{x^2} + C_2 x$$