# NonHomogeneous Second Order using Undetermined Coefficients

1. Sep 30, 2011

### Lucci

Hi all,
I understand the basic concept of undetermined coefficients, but am a little lost when g(t) in the equation yll+p(t)yl+q(t)y=g(t) is a product of two functions. The specific problem I'm working on is as follows:

yll-2yl-3y=-3te-t

When I solve for the homogeneous set of solutions I get roots 3 and -1
(r2-2r-3)=0
(r-3)(r+1)=0
Therefore, I have y(t)=c1e-t+c2e3t

Now, if g(t) were just equal to -3e-t I would just set Y(t)=Ae-t and use Y(t) to solve for the particular solution.

However, because g(t) is the product of two equations, I am not sure how to proceed at this point. Someone suggested that I use the homogeneous set of solutions as my Y(t), solve for Yl(t) and Yll(t) and plug those back into my original equation. Is this the correct way to approach this problem? And if so, how exactly am I supposed to do this?

Thanks for any help!!

2. Oct 1, 2011

### AuraCrystal

Hey,

For your particular solution, normally you would use
$y_{p}(x)=(At+B)e^{-t}$

where A and B are to be determined. However, $e^{-t}$ is a term in the homogeneous solution so you need to multiply the particular solution by t so that the problem is fixed. You then plug it into the equation and proceed as normally.

Hope that helps!

Also, Boyce and DiPrima gives a nice discussion of this stuff! ^_^

Last edited: Oct 1, 2011