# solve the differential equation by undetermined coefficients

• Mar 15th 2011, 10:48 AM
slapmaxwell1
solve the differential equation by undetermined coefficients
y'''-3y''+3y'-y=x-4e^x

ok so i found the roots of the equation where m = 1 and then i got my yc equation which was also pretty straight forward.

what i am having problems with is the yp equation,

i made my guess to be: Ax + Bx^3e^x <--- my book said i was wrong?

any help would be appreciated...
• Mar 15th 2011, 11:33 AM
dwsmith
Quote:

Originally Posted by slapmaxwell1
y'''-3y''+3y'-y=x-4e^x

ok so i found the roots of the equation where m = 1 and then i got my yc equation which was also pretty straight forward.

what i am having problems with is the yp equation,

i made my guess to be: Ax + Bx^3e^x <--- my book said i was wrong?

any help would be appreciated...

I think it should be just $Ax+Be^x$
• Mar 15th 2011, 11:41 AM
Ackbeet
Quote:

Originally Posted by dwsmith
I think it should be just $Ax+Be^x$

When you plug that into the LHS, the operator will annihilate the exponential, so there's no way you'll get the RHS out of that. Because the root m = 1 has multiplicity 3, your particular solution will have to look something like slapmaxwell1's candidate. I might recommend have something more like

$y_{p}=A+Bx+Ce^{x}+Dxe^{x}+Ex^{2}e^{x}+Fx^{3}e^{x}.$

That might be overkill, but it might not. A hunch: any of those constant that are superfluous will be easy to solve for, I think. So I don't think you'll waste too much time doing it this way.