Solve x²y''+xy'−4y=x²

x²y'' + xy' − 4y = x² is a Cauchy–Euler equation.

Transformation

Put z = log x, D = d/dz:

[D(D − 1) + D − 4]y = e^(2z) (D² − 4)y = e^(2z)

Complementary function

m = ±2: CF = c₁e^(2z) + c₂e^(−2z) = c₁x² + c₂x⁻²

Particular integral

Since 2 is a root of D² − 4 = 0, use the resonance rule (derivative of the denominator is 2D = 4):

PI = z e^(2z)/(2·2) = (x² log x)/4

General solution

y = c₁x² + c₂/x² + (x² log x)/4