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What can I guess as a particular solution for $y''+25y = -x\sin(5x)$

I tried $(Ax+B)(C\cos(5x)+Dsin(5x))$, but that didn't work.

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That's the 5th (!) question by you within an hour or so, and they're all very similar. Please, just google for a table of common guesses for solutions of linear ODEs with various error terms –  fgp Oct 20 '12 at 1:05
    
i.imgur.com/9OzUf.jpg –  user1038665 Oct 20 '12 at 1:06
    
Here's one such table: en.wikipedia.org/wiki/Method_of_undetermined_coefficients –  fgp Oct 20 '12 at 1:06
    
That's what I used. I don't know why my solution is incorrect, though. –  user1038665 Oct 20 '12 at 1:08
    
Then, at the very least, explain what didn't work. Put the actual equation you got for the coefficients into your question. –  fgp Oct 20 '12 at 1:11
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2 Answers

up vote 1 down vote accepted

We need to use something similar to the right hand side i.e. $${y_p} = Ax\sin (5x) + Bx\cos (5x).$$

Do you understand why?

If you do not understand why this is the way we guess, then try another but longer method. Take all the functions that appear in the product on the right hand side. Multiply them together and then simply put undetermined coefficients in from of each term.

If one of the functions is $\cos (ax)$ or $\sin (bx)$ then $\cos (ax) + \sin (bx)$ should appear in the product.

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Suppose the right hand side were ${x^4}{e^{2x}}\cos (3x)$. What would be your guess? –  glebovg Oct 20 '12 at 1:47
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Here is a list of different combinations of functions for particular solutions.

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