Solve (D²−D−2)y=eˣ+sin 2x

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

Complementary function

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

CF = c₁e^(2x) + c₂e^(−x)

Particular integral

For eˣ: put D = 1: 1/(1 − 1 − 2) eˣ = −eˣ/2.

For sin 2x: put D² = −4:

1/(D² − D − 2) sin 2x = 1/(−6 − D) sin 2x = −1/(D + 6) sin 2x = −(D − 6)/(D² − 36) sin 2x

With D² = −4 the denominator is −40:

= −(D − 6)/(−40) sin 2x = (D − 6) sin 2x/40 = (2 cos 2x − 6 sin 2x)/40 = −(3 sin 2x − cos 2x)/20

General solution

y = c₁e^(2x) + c₂e^(−x) − eˣ/2 − (3 sin 2x − cos 2x)/20