Word Count : 448

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We are asked to solve the differential equation: [ x^2 y'' - 2x y' - 4y = x^2 + 2 \ln x ] Step 1: Solve the corresponding homogeneous equation [ x^2 y'' - 2x y' - 4y = 0 ] This is a Cauchy-Euler (equidimensional) equation. We assume a solution of the form (y = x^m). Then [ y' = m x^{m-1}, \quad y'' = m(m-1)x^{m-2} ] Substitute into the homogeneous equation: [ ____ __________ _______ ____ _____ _____.
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