2nd order non-homogeneous solution help needed

tremor

New member
Joined
Sep 22, 2010
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19
hi everyone,

i have a solution to this problem but i'm not sure if it's correct. there is a lot of work to show so i will do my best to summarize. i believe somewhere in the steps i went wrong. i would very much appreciate another set of eyes!
d^2y/dx^2-4dy/dx + 4y = (1+x+x^2+x^3)e^(2x)
with y(0) = 1; dy(0)/dx=1
i started by finding yp and then yh.
yp=(Ax^2+Bx+C)e^2x
y'p=(2Ax +B)e^2x; y''p=(2A)4e^2x
yp''-4yp'+4yp=1+x+x^2+x^3)e^(2x)
i substitute in those values and solve for A, B, & C. the e term cancels out ( i think!)
i get yp=1/4x^2+5/4x+11/4
i get yh=C1e^2x+C2+xe^2x
so y = yp+yh
initial conditions allow me to solve so C1=-7/4 and C2=1/2
y = -7/4e^2x+1/2+xe^2x + 1/4x^2+5/4x+11/4

am i on the right track?

thanks!!
 
Your homogeneous solution is correct. I didn't check the particular solution, but you are certainly on the right track.
 
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