Solve (D²+2D+1)y=eˣ+x²

The equation is (D² + 2D + 1)y = eˣ + x².

Complementary function

Auxiliary equation m² + 2m + 1 = 0, so (m + 1)² = 0, m = −1, −1.

CF = (c₁ + c₂x)e^(−x)

Particular integral

For eˣ: put D = 1:

PI₁ = eˣ/(1 + 2 + 1) = eˣ/4

For x²:

PI₂ = 1/(1 + D)² · x² = (1 + D)⁻² x² = (1 − 2D + 3D² − …) x² = x² − 2(2x) + 3(2) = x² − 4x + 6

General solution

y = (c₁ + c₂x)e^(−x) + eˣ/4 + x² − 4x + 6