is typed as lambda, as alpha.

Assume the PDE is separable by making the substitution :
Move the term with to the right hand side and divide both sides by so that we get the separated ODE's:
= =

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).

For this set of differential equations we have two bifurcations, whether in the case of X or in the case of Y.

Case 1:

Case 2:

Case 3:

Case 3:

Case 3:

You can earn partial credit on this problem.