Solve using Undetermined Coefficients

Question

(D^2+2D+1)y = sin x + 3 cos 2x

Answer 1

y'' + 2y' + y = 0; Characteristic equation: r² + 2r + 1 = 0; r = - 1 multiplicity 2.

y_HG = (Ax + B)e^-x; y_P = Msinx + Ncosx + Kcos2x + Lsin2x; y_P' = Mcosx - Nsinx - 2Ksin2x + 2Lcos2x;

y_P'' = - Msinx - Ncosx - 4kcos2x - 4Lsin2x;;

- Msinx - Ncosx - 4Kcos2x - 4Lsin2x; + 2Mcosx - 2Nsinx - 4Ksin2x + 4Lcos2x + Msinx + Ncosx + Kcos2x +

Lsin2x = sinx + 3cos2x; - 3Kcos2x + 4Lcos2x - 3Lsin2x - 4Ksin2x + 2Mcosx - 2Nsinx = sinx - 3cosx;

-3K + 4L = 3; - 3L - 4K = 0; 2M = 0, - 2N = 1; M = 0; N = - 1/2; K = - 9/25; L = 12/25

y + (AX + B)e^-x + - 1/2cosx - 9/25cos2x + 12/25sin2x