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May 2nd, 2015, 05:57 PM   #1
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Using Separation of variables

Hello everyone,

I am having an issue with this concept. I am running into having a second order differential equation when I am using the separation of variables method. I would normally type the problem into this text space but I figured it was better for me to upload a picture created from microsoft's word processor with the formatting of equations.

Please see attached jpeg for problem.

So since I have a second order and there are three different ways for me to solve a second order, would the general solution of the second order include a summation to cover for all three cases?
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Last edited by philm; May 2nd, 2015 at 06:00 PM. Reason: Forgot to upload document
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May 2nd, 2015, 08:51 PM   #2
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Yes, I believe that the general solution is the sum of all distinct solutions. Assuming all different methods give you different solutions, your answer would be a sum of all those solutions.
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May 3rd, 2015, 03:48 PM   #3
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Hm, intersting, There are three different cases depending on the complex roots of the 2nd order characteristic equation. This is all good and everything but the lambda is causing a huge issue since it is unknown. After working out the complex roots, I determined the three cases (w = lambda)

1) w = 1/4
2) w > 1/4
3) w < 1/4

I am not too sure but I think that there is a particular case that I have to choose from for this specific problem? Or do I have to solve the PDE for all three cases?
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May 5th, 2015, 03:38 PM   #4
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I was able to solve this problem
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May 6th, 2015, 05:37 AM   #5
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Quote:
Originally Posted by philm View Post
I was able to solve this problem
Great! Did you need to solve all three cases?
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May 6th, 2015, 10:41 AM   #6
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I had to take the summation of the separation of variables.

Then, u(x, t) only existed at one value, n = 1.

At that point, lambda went away. It was a little tricky this one.

Then I solved the 2nd order ODE using IC and BC.
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