(1 pt) is typed as lambda, a as alpha. The PDE ди ди k² y5 дх² ду is separable, so we look for solutions of the form u(x,t) = X(x)Y(y). 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 X"/(X) =Y'/(k^2y^6) =-λ Note: Use the prime notation for derivatives, so the derivative of X is written as X'. Do NOT use X' (x) Since these differential equations are independent of each other, they can be separated DE in X: DE in Y: = 0 = 0 Now we solve the separate separated ODES for the different cases in 1. 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:= 0 X(x) = Y(y) = DE in Y Ifλ0, 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 y (placing the constant of integration c in the right hand side) we get Solving for Y, using the funny algebra of constant where e = c is just another constant we get Y = For 0 we get a Sturm-Louiville problem in X which we need to handle two more cases Case 2: λ = -a² X(x) = Case 3:λ = a² X(x) = Final Solution Case 1:λ=0 u = Case 2: a² u = Case 3:λ=a² u =