Variation of Parameters (Integral Trouble)

[Saladsamurai]

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

 

[Saladsamurai]

Does what I have so far look correct?

Any advice before I proceed is appreciated.

 

[HallsofIvy]

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.

 

[Saladsamurai]

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

 

[Saladsamurai]

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!

 

[Saladsamurai]

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.

 

[HallsofIvy]

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".

 

[Saladsamurai]

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".

Right I am assuming that Xp takes the form of F(t) and all possible derivatives of it.

 

[Saladsamurai]

Method of undetermined coefficients + http://www.gregthatcher.com/Mathematics/GaussJordan.aspx" =