NonHomogeneous Second Order using Undetermined Coefficients

[Lucci]

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

y^ll-2y^l-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+c₂e^3t

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 Y^l(t) and Y^ll(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!

 

[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! ^_^