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Question 05 (a) 

Consider the following problem involving Laplace's equation in an annular region:
Use the method of separation of variables to solve the problem when 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Similar to question 2b of this exam, when using the method of separation of variables, we would like to write as a product of two solutions of a single variable. In question 2b, we wrote as but we need something slightly different this time (because there is no or anywhere in the equation!). How should we write as a product of two solutions of a single variable this time? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To use the method of separation of variables, we write as We compute that and Substitute the above into our differential equation we get that Next, we would like to group all the terms onto one side and all the terms onto another, so divide the entire equation by to get which gives To get rid of the factor of in the last term on the left, we will multiply the equation by to get which gives Here, and are independent of each other. However, the right hand side of the equation is a function of while the right hand side a function of , so both sides of the equation must be equal to a constant which we will denote . In other words, we have This gives us two equations We will first solve the equation for . Before we go ahead to do so, we would like to figure out the boundary conditions for . The boundary conditions imply that Since we do not want to be since this will give the trivial solution (i.e. ) which is not of interest to us. So the boundary conditions give us We will now solve the equation for which says This is the Euler's equation and we recall that we should look for solutions in the form for an unknown constant to be determined. We compute that Substituting these into we get that which after simplifying gives Here, the sign of will affect the form of the solution, so we have three cases to consider. They are
Further calculations show that case 2 and case 3 only give trivial solutions. If you are not sure, you should try verifying that yourself. For completeness of the solution, I will also include the computation in the appendix of this solution. Now, case 1: says that This gives Recall that for the Euler's equation, if is in the form of , the solution of the Euler's equation can be written as for some arbitrary constants and . For us, we have and , so our solution for is Next, we need to match the boundary conditions Hence, and Now, If , we will have the trivial solution. To get nontrivial solutions, we need In other words, we have an infinite numbers of eigenvalues give by with the corresponding eigenfunctions Now, we will go back to solve the equation. Recall that we have so the equation becomes which gives solutions of the form for some arbitrary constants and . Since we found that an infinite number of , we have an infinite number of given by Putting everything together, we can express as We can absorb the arbitrary constant into and to get Finally, we will use the other two boundary conditions and to find and . Hence Finally, to find , we use the last boundary condition
This boundary condition gives us that If we expand out the summation, we see that Here, we see that is the only term on the left hand side that matches with the right hand side. Hence, we have and In other words, the solution is
We will address two final questions in this appendix. Question 1: Refer to the solution above, why doesn't lead to nontrivial solutions for the Euler's equation? Answer to question 1: Suppose , then The solution to the Euler's equation is for some arbitrary constants and . The boundary condition implies So The boundary condition implies Hence, we only get the trivial solution.
Answer to question 2: Suppose , then The solution to the Euler's equation is for some arbitrary constants and . The boundary condition implies So The boundary condition implies If , we get a trivial solution, so suppose , then we must have However, this contradicts with our original assumption that . 