solution of differential equation - (D^2 - 2D + 4)y = e^2x cos x.

The Auxiliary equation is,                                                                          

m² - 2m + 4 = 0

m = [2 ± √(4 – 16)] ∕ 2

m = 1 ± 2√3i/3

m = 1± √3i

complementary function = eˣ [A cos √3x + B sin√3x]

                                         = Aeˣ [cos √3x+B]

Particular integral=1/(D²-2D+4) e²ˣ cos x

                            = e²ˣ 1/[(D+2)²-2(D+2)+4] cos x

                            = e²ˣ 1/(D²+2D+4) cos x

                            = e²ˣ 1/(-1+2D+4) cos x

                            = e²ˣ 1/(2D+3)*(2D-3)/(2D-3) cos x

                            = e²ˣ 1/(4D²-9) 2 sin x- 3cos x

                            = -e²ˣ/13 [2 sin x- 3cos x]

y = complementary function + particular integral

y = eˣ [ A cos √3x + B sin √3x] - e²ˣ/13 [2 sin x – 3 cos x]