Solve (D²−2D+5)y=10 sin x

The equation is (D² − 2D + 5)y = 10 sin x.

Complementary function

m² − 2m + 5 = 0, so m = 1 ± 2i.

CF = eˣ(c₁ cos 2x + c₂ sin 2x)

Particular integral

Put D² = −1:

PI = 10/(D² − 2D + 5) · sin x = 10/(4 − 2D) · sin x = 10(4 + 2D)/(16 − 4D²) · sin x

With D² = −1 the denominator is 20:

= (4 + 2D) sin x / 2 = (4 sin x + 2 cos x)/2 = 2 sin x + cos x

General solution

y = eˣ(c₁ cos 2x + c₂ sin 2x) + 2 sin x + cos x