is typed as lambda, as alpha.

The PDE is separable, so we look for solutions of the form . When solving DE in X and Y use the constants a and b for X and c for Y.
The PDE can be rewritten using this solution as (placing constants in the DE for Y) into
= =

Note: Use the prime notation for derivatives, so the derivative of is written as . Do NOT use

Since these differential equations are independent of each other, they can be separated
DE in X:
DE in Y:

Now we solve the separate separated ODEs for the different cases in . In each case the general solution in X is written with constants a and b and the general solution in Y is written with constants c and d. Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be acos(x) + bsin(x).
Case 1:

DE in Y
If , the differential equation in Y is first order, linear, and more importantly separable. We separate the two sides as
Integrating both sides with respect to (placing the constant of integration c in the right hand side) we get =
Solving for Y, using the funny algebra of constant where is just another constant we get

For we get a Sturm-Louiville problem in X which we need to handle two more cases
Case 2:

Case 3:

Final Solution
Case 1:

Case 2:

Case 3:

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