Question

find the particular solution

what do I do when the homogeneous equation is not factorable?

Answer 1

"Y"" + 3y' - 2y = -e^2x (9 sin 2x + 5 cos 2x) y(x) = y_c (x) + y_p (x) particular solution corresponding homogeneous equation y"" + 3y' - 2y = 0 characteristic equation is given by m^2 + 3m - 2 = 0 implies m = -3 plusminus squareroot 9 + 8/2 = -3 plusminus squareroot 17/2 Therefore Fundamental solution set = { e^-3 plusminus squareroot 17/2 x, e^-3 -squareroot 17/2} (We will find y_p(x) by method of undetermined coefficient) Undetermined coefficient set {e^2x sin 2x, e^2x cos 2x yp' = 2 A e^2x sin 2x + 2 A e^2x cos 2x + 2 B e^2x cos 2x - 2 Be^2x + sin 2x yp' = (2A - 2B) e^2x sin 2x + 2(2A - 2B) e^2x cos 2x + 2(2A + 2B) e^2x cos 2x - 2(2A + 2B) e^2x sin 2x = (4A - 4B - 4A - 4B) e^2x sin 2x + (4A - 4B + 4A + 4B) e^2x cos 2x yp"" = -8 B e^2x sin 2x + 8 A e^2x cos 2x putting yp, yp', yp"" in * and comparing coefficient."