Variation of Parameters (Integral Trouble)

In summary, the Differential Equation has a star next to it and the person is trying to integrate it but they don't know how to use variation of parameters. They think that using tabular integration will be faster than doing it by parts, but they are wrong.
  • #1
Saladsamurai
3,020
7
So I pretty much have this Differential Equation solved except that I have to integrate the expression [itex]\int \Phi(t)F(t)dt[/itex] 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?

Picture1-9.png


Thank you
 
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  • #2
Does what I have so far look correct?

Any advice before I proceed is appreciated.
 
  • #3
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.
 
  • #4
HallsofIvy said:
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
 
  • #5
Actually, I could use tabular integration on these since the terms with t^2 are of the form
[itex]\int p(t)f(t)dt[/itex] ... 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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  • #6
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.
 
  • #7
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".
 
  • #8
HallsofIvy said:
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.
 
  • #9
Method of undetermined coefficients + http://www.gregthatcher.com/Mathematics/GaussJordan.aspx" =:smile:
 
Last edited by a moderator:

1. What is the Variation of Parameters method?

The Variation of Parameters method is a technique used to find a particular solution to a non-homogeneous linear differential equation. It involves finding a set of functions that satisfy the homogeneous equation, and then using them to construct a particular solution.

2. How is the Variation of Parameters method different from other methods of solving differential equations?

The Variation of Parameters method is different from other methods because it does not rely on a predetermined formula or algorithm. Instead, it involves finding a set of functions that satisfy the homogeneous equation, which allows for a more flexible and adaptable approach to finding a particular solution.

3. When is the Variation of Parameters method most useful?

The Variation of Parameters method is most useful when the non-homogeneous term in a differential equation is a function that cannot be easily integrated, such as polynomials or trigonometric functions. In these cases, the method allows for a more straightforward and efficient way to find a particular solution.

4. What are the steps involved in using the Variation of Parameters method?

The steps for using the Variation of Parameters method are as follows:

  • 1. Solve the homogeneous equation to find a set of linearly independent solutions.
  • 2. Use these solutions to construct the Wronskian, a determinant that will be used in the next step.
  • 3. Set up a system of equations using the Wronskian and the non-homogeneous term.
  • 4. Solve the system of equations to find the coefficients of the particular solution.
  • 5. Add the particular solution to the general solution of the homogeneous equation to get the complete solution.

5. Can the Variation of Parameters method be used for higher-order differential equations?

Yes, the Variation of Parameters method can be extended to higher-order differential equations. The process is the same, except that the Wronskian and the system of equations will involve more functions and coefficients.

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