Give the form of a particular solution of y4 6 y 10 y 6 y

Give the form of a particular solution of y^(4) + 6 y\'\'\' + 10 y\" + 6 y\' + 9 y = 4e^-3x + cos (3x) + 3 given that r_1 = 1i is a root of the characteristic equation. z = Ae^-3x + Bx cos(x) + Cx sin(x) + D z = Ae^-3x + B cos(3x) + C sin(3x) + D z = Axe^-3x + B cos(3x) + C sin(3x) + D z = Ax^2e^-3x + Bx cos(x) + Cx sin(x) + D z = Ax^2e^-3x + B cos(3x) + C sin(3x) + D None of the above.

Solution

We make a guess of particular solution based on the inhomogeneous part

THe guess for e^{-3x} is Ae^{-3x}

FOr cos(3x) is B cos(3x)+C sin(3x)

For 3 it is D

So, the form of particular solution is

A e^{-3x}+B cos(3x)+C sin(3x)+D

ie option (b)