Solve (D²−6D+9)y=x²e^(2x)

The equation is (D² − 6D + 9)y = x²e^(2x).

Complementary function

m² − 6m + 9 = 0, so (m − 3)² = 0, m = 3, 3.

CF = (c₁ + c₂x)e^(3x)

Particular integral

Use the shift rule, replacing D by D + 2:

PI = e^(2x) · 1/[(D + 2 − 3)²] · x² = e^(2x) · 1/(D − 1)² · x² = e^(2x)(1 − D)⁻² x² = e^(2x)(1 + 2D + 3D² + …) x² = e^(2x)(x² + 4x + 6)

General solution

y = (c₁ + c₂x)e^(3x) + e^(2x)(x² + 4x + 6)