Question

Solve the equation (D2 - 4D + 3) y = sin 3x + x2.​

Answer 1

Answer:

For the differential equation y’’ - 4y’ + 3y = sin3x + x^2 the characteristic equation is

m^2 - 4m + 3 =0 with roots 1 , 3.Then the solution of the homogeneous part is

yh = C1e^x + C2e^3x. For the particular integral let’s try the practical approach

assuming yp = Asin3x + Bcos3x + Cx^2 + Dx +E. Substituting in the equation leads to

A = -1/30 , B = 1/15 , C = 1/3 , D = 8/9 , E =26/27, Then the solution of the equation

is y = C1e^x + C2e^3x +(1/15)cos3x - (1/30)sin3x +(1/3)x^2 + (8/9)x +26/27.