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- (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 + bx? (5) Write differential equations with leading term positive, so X" – 2X = 0 rather than - X" + 2X = 0. The PDE a2u a2u ax2 at2 is separable, so we look for solutions of the form u(x, t) = X(x)T(t) The PDE can be rewritten using this solution as (placing constants in the DE for T) into X"/X = (-1/^2)*(T"/Y) = -2 Note: Use the prime notation for derivatives, so the derivative of X is written as X'. Do NOT use X'(x)

Answer 1

geven that, ažu = K2 2 2 2+2 D het un, t) be the solution of i'. M2H) = x () T(1) and substibeting into equation ③ equation ☺ , we obtain хат K² T dx2 2 at 2 => dx/da2 d²T/ dt? 2 = -2, X => X" 1] x K I" X :. xn x F-1 Il = -2x => X" x" + ax = 0, thés és DE in X the and, t T" T --) K2 2 27 T 11 ATK? 2 >>T +kXT =0 this is the ein T. Case I a = 0 . x" = 0 ने x'a a => X => x = axtb :.*(2) = axtb) where a b centlant.

wehere q c2, and C4 and, T"=0 > T'=c * T = ettd ist (+) >Cd + d where cod are arbitray caustents. ilwer, t) = (ax+b) (e++d) case a 7=-2² is X" -&? x auxiliary ean . eans, m²_x²=0 29 m =Id. ... *(a) = geta of geda T" + k” (-2) T0 T' - K²L?T=0 ausiliony egna, m²-k²a m 2 - o 7 m fka T(A) ze kat e - Kat 3 of ca e (a parte e ez eda] (a .kat are constants.

Case II 2=2² x" +22 X=0 ausiliary equation ei, m² +2²= 27 m £ cu ..|*(x)=(d, cosaf da sinda) = 0 and T"tk²2² T=0 alexiliary equation is, m² +622²=0 1 7 ma I Kai ..h(+) = (da cos(kx st) + da sin (kat) clues,t)=(che cosaa of de fin da a) (da cos(kat) + da sin (kat) where da, dz, d,, da are castante