# Variation of Parameters (Integral Trouble)

So I pretty much have this Differential Equation solved except that I have to integrate the expression $\int \Phi(t)F(t)dt$ it has a star next to it in my attached work.

Does this look readily integrable to anyone? For some reason nothing is ringing a bell. I suppose I could go by parts, but I have a feeling that will suck even worse than the work up to this point.

I hope I didn't make any errors up until this point. Does this look reasonable?

Thank you

Does what I have so far look correct?

Any advice before I proceed is appreciated.

Homework Helper
I see no problem with integrating it. Integration by parts, twice, should take care of the terms with t2 in them. Are you required to use "variation of parameters"? This looks like "undetermined coefficients" would be much easier.

I see no problem with integrating it. Integration by parts, twice, should take care of the terms with t2 in them. Are you required to use "variation of parameters"? This looks like "undetermined coefficients" would be much easier.

Really? See I had thought that Variation would be easier since there was a t^2 term. So Xp would take the form all of those derivatives. Then I would have 8 unknowns and eight equations.

But really, I have done this entire problem using undetermined coefficients. I thought I should at least show that I know how to use Variation of parameters.

So back on track. Integrate by parts, and then multiply that result by the fundamental matrix and I should have Xp.

Thanks

Actually, I could use tabular integration on these since the terms with t^2 are of the form
$\int p(t)f(t)dt$ ... I think that will be quicker than by parts.

EDIT: This is stupid!!!! This leads me to wonder why Variation of parameters would EVER be easier??? After all of this work, I am going to switch to "Undetermined coefficients" .... There are too many places to make algebraic errors doing it by parameters. This integration is a mess!

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I am going to stab someone. After all of this (I finished it by Variation..) I plugged back into my original DE and it doesn't work out. Just thought I'd share my pain with you.

Homework Helper
I'm going to be watching my back for a while!

If t2 is on the right side of a linear differential equation with constant coefficients, you will need to try something like "At2+ Bt+ C".