Solve (D²+2D+1)y=eˣ cos x

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

Complementary function

m² + 2m + 1 = 0, so m = −1, −1.

CF = (c₁ + c₂x)e^(−x)

Particular integral

Replace D by D + 1 and take eˣ outside:

PI = eˣ · 1/[(D + 1)² + 2(D + 1) + 1] · cos x = eˣ · 1/(D² + 4D + 4) · cos x

Put D² = −1: denominator 4D + 3.

= eˣ · (3 − 4D)/(9 − 16D²) · cos x

With D² = −1, 9 − 16D² = 25:

= (eˣ/25)(3 cos x − 4D cos x) = (eˣ/25)(3 cos x + 4 sin x)

General solution

y = (c₁ + c₂x)e^(−x) + (eˣ/25)(3 cos x + 4 sin x)