Question

Important Instructions: (1) λ is typed as lambda. (2) Use hyperbolic trig functions cosh(x) and sinh(x) instead of ex and e−x. (3) Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be Acos(x)+Bsin(x). (4) For polynomials use arbitrary constants in alphabetical order starting with highest power of x, for example, Ax2+Bx. (5) Write differential equations with leading term positive, so X′′−2X=0 rather than −X′′+2X=0. (6) Finally you need to simplify arbitrary constants. For example if A, B and C are arbitrary constants then AC, BC could be simplified to A, B since C in this case would be redundant. The PDE k2∂2u∂x2+∂2u∂y2=0 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 and D for Y. The PDE can be rewritten using this solution as (place constant k in the DE for Y) into X''/X = = −λ Note: Use the prime notation for derivatives, so the derivative of X or Y is written as X′ or Y′. 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: =0 (your answer should be simplified so there are no fractions) Now we solve the separated ODEs for the different cases in λ. 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 A cos(x) + B sin(x). On this exercise you are required to keep all arbitrary constants in your responses. Case 1: λ=0 X(x)= Y(y)= u(x,y)= Case 2: λ=−a2 For λ=−a2 we get two second order linear problems, one in X and one in Y. X(x)= Y(y)= u(x,y)= Case 3: λ=a2 X(x)= Y(y)= u(x,y)=

Answer 1