Solve the differential equation: 

To solve the given differential equation \(x^2 \frac{d^2y}{dx} - 4y = x^2 + 2 \ln(x)\), we can use the method of undetermined coefficients. 

First, let's solve the associated homogeneous equation:

\[x^2 \frac{d^2y_h}{dx^2} - 4y_h = 0\]

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