To solve the given differential equation

$x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} - 12y = x^3 \log(x),$

we can follow these steps:

1. Identify the type of differential equation: This is a second-order linear differential equation with non-constant coefficients.

2. Solve the homogeneous equation: First, we solve the corresponding homogeneous equation

$x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} - 12y = 0.$