Question

please show all work neatly! also this is a differential eqaution problem

Answer 1

"Given y^"" - 3y^' = 8e^3x + 4 sinx (1) or (D^2 - 3D)y = 8e^3x + 4 sinx the auxiliary eq^2 is D^2 - 3D = 0 = > D(D - 3) = 0 = > D = 0, 3 Therefore c.f = c_1 e^0x + c_2 e^3x = c_1 + c_2 e^3x since the term e^3x in the R.H.S of (1) appear in c.f Therefore corresponding to e^3n of R.H.S of (1) we choose trial solution for p.I as A_1 xe^3x and corresponding to decimal form sinx of R. H.S of (1) we choose trial solution for P.I as A_2 cosx + A_3 sinx combining These we get a trial solution for P.I of the form y^* = A_1 x e^3x + A_2 cosx + A_3 sinx (2) since y^* must satisfy (1) D^2 y^* - 3Dy^* = 8e^3x + 4 sinx -(3) from(2) DY^* = 3A_1xe^3x + A_1e^3x - A_2 sinx + A_3 cosx (4) DY^* = 3A_1xe^3x"