y^(4) - 5y'' + 4y = e^x - xe^2x
Find the general solution yp of the given equation
Solve the homogeneous equation
your soln's are e^2x, e^(-2x) , e^x and e^(-x )
for yp for the e^x term you'd normally have an Ae^x term but it is a homog solution so use Ax*e^x
For x^e(2x) normally you would have a Bxe^(2x) and a Ce^(2x)
but since e^(2x) is a solution to the homogeneous eqn
multiply each by x so you'll have Bx^2e^(2x) and Cxe^(2x)
yp = Axe^x + Bx^2*e^(2x) + Cx*e^(2x)
Now you're ready to go to work