Use the method of undetermined coefficients to solve the differential equation y” + 4y’ + 4y = cos x + e^-3x /2. Help me please, thank you sir.

Question

Use the method of undetermined coefficients to solve the differential equation y” + 4y’ + 4y = cos x + e^-3x /2. Help me please, thank you sir.

Answer 1

k² + 4k + 4 = 0; k = - 2 and general solution of homogenius equation y_GH = (Cx + D)e^-2x;

Particular solution of given ODE: y_P = .Asinx + Bcosx + Ee^-3x/2;

y'_P = Acosx - Bsinx - 3/2Ee^-3x/2; y''_P = - Asinx - Bcosx + 9/4Ee^-3x/2;

-Asinx - Bcosx + 9/4Ee^-3x/2 + 4Acosx - 4Bsinx - 6Ee^-3x/2 + 4Asinx + 4Bcosx + 4Ee^-3x/2 = cosx + e^-3x/2;

3A - 4B = 0; 3B + 4A = 1; E/4 = 1 ; E = 4; A = 4/25; B = 3/25.

y_P = 4sinx/25 + 3cosx/25 + 4e^-3x/2.

General solution of given ODE: y = (Cx + D)e^-2x + 4sinx/25 + 3cosx/25 + 4e^-3x/2.