Question

Given:

y''+2y'=2x+5-e^-2x

General solution is: y=c1e^-2x+c2 +1/2(x^2)+2x+1/2(xe^-2x)

Solve using the method of undetermined coefficients and show all steps please!
I have the form of yp is Ax^2+Bx+Cxe^-2x, and the issue that plagues me is in solving for A B C. I get A=1/2 and I get B=2, but the terms involving C fall off the face of the earth when I substitute y' and y'' of the solution form into the equation, so how can I solve for C? Help Please and Thanks!!!!

Answer 1

The auxiliary equation is with roots r=0 and r=-2, so the solution of the complementary is . For the equation we try

            

So substitution in the equation gives

Comparing the coefficients to get the system of equations

The general solution is