
Differential Equations Ordinary and Partial Differential Equations Math Forum 
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May 2nd, 2015, 06:57 PM  #1 
Newbie Joined: Sep 2014 From: n/a Posts: 9 Thanks: 0  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? Last edited by philm; May 2nd, 2015 at 07:00 PM. Reason: Forgot to upload document 
May 2nd, 2015, 09:51 PM  #2 
Senior Member Joined: Aug 2014 From: United States Posts: 136 Thanks: 21 Math Focus: Learning 
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.

May 3rd, 2015, 04:48 PM  #3 
Newbie Joined: Sep 2014 From: n/a Posts: 9 Thanks: 0 
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? 
May 5th, 2015, 04:38 PM  #4 
Newbie Joined: Sep 2014 From: n/a Posts: 9 Thanks: 0 
I was able to solve this problem

May 6th, 2015, 06:37 AM  #5 
Senior Member Joined: Aug 2014 From: United States Posts: 136 Thanks: 21 Math Focus: Learning  
May 6th, 2015, 11:41 AM  #6 
Newbie Joined: Sep 2014 From: n/a Posts: 9 Thanks: 0 
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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separation, seperation, variables 
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