Notice that the left hand side is a derivative of a product:
$xy'' + y' = \frac{d}{dx}(x y') = 4x$.
Integrating both sides with respect to $x$:
$x y' = \int 4x\,dx = 2x^2 + C_1$.
Dividing by $x$:
$y' = \frac{dy}{dx} = 2x + \frac{C_1}{x}$.
Integrating once more with respect to $x$:
$y = \int \left(2x + \frac{C_1}{x}\right)dx = x^2 + C_1 \ln x + C_2$.

Detailed analysis of options:

• $y = x^3 + C_1 \ln x + C_2$ has $x^3$ instead of $x^2$.
• $y = x^2 + C_1 \ln x + C_2$ is correct.
• $y = e^x + C_1 \ln x + C_2$ is incorrect.
• $y = e^{-x} + C_1 \ln x + C_2$ is incorrect.