Question

(7 points) Important Instructions: (1) is typed as lambda and a is typed as alpha. (2) Use hyperbolic trig functions cosh(x) and sinh(x) instead of et and e-2. (3) Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be a cos(x) + b sin(x). (4) For polynomials use arbitrary constants in alphabetical order starting with lowest power of x, for example, ax + bx2. (5) Write differential equations with leading term positive, so X" – 2X = 0 rather than – X" + 2X = 0. The PDE • a²u 22 au at2 ar2 is separable, so we look for solutions of the form u(x, t) X(xT(t) The PDE can be rewritten using this solution as (placing constants in the DE for T) into = -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: 0 DE in T: =0

Answer 1

Answer zu k² 2²u dX Tit) 1) Xa dT at dt and õu d²x T(H) dt² an2 2t² let ulegt) X (2) Tt) =) 3 a zu X w d²T da² an dt² 2²u Putting values of gen in equation Jt² d²x T(t) da² dt² 1² dx d²T X (a) du² Tt) dt2 equation is possible if 2x² k2 X (2) d²T 11 k2 Constant Xa) da² TE) It² = – A Take Constant dex d x² k 2 d²I Tt dt² -1 X (n)

TH) Gi Cos (at) + Sin (& t) So, u(at) (Fi CACE") + sin(n)) (4, 684€ 778e siner t)