Question

(2 points) is typed as lambda, a as alpha. The PDE a2u ar2 = yº ди ay 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 (1/(k^2))(y^5)(Y'/Y) = -2 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: X"+lambdaX 0 DE in Y lambdak2Y+y^5Y' = 0 с 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: 1=0 X(2) = a+bx Y(y) = DE in Y If I = 0, the differential equation in Y is first order, linear, and more importantly separable. We separate the two sides as Y/Y -((k^2)/(y^5))lambda Integrating both sides with respect to y placing the constant of integration c in the right hand side) we get In(absY) Solving for Y, using the funny algebra of constant where ec = c is just another constant we get Y = For 1 + 0 we get a Sturm-Louiville problem in X which we need to handle two more cases Case 2: = -a? X(X) = a*cosh(alphax)+b*sinh(alphax)

Answer 1

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