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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

(r^{2}-2r-3)=0

(r-3)(r+1)=0

Therefore, I have y(t)=c_{1}e^{-t}+c_{2}e^{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!!

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# NonHomogeneous Second Order using Undetermined Coefficients

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