Question

(1 poin This problem is concerned with using separation of variables to find product solutions. In particular you will substitute ( separate the variables. Then let - represent the separation constant. Solve the resulting ODEs and find (x,1). 1) X() into the given equation and Use separation of variables to find product solutions of the partial differential equation. Separation of variables gives - P T ' + p = 0, The general solution of T''+pT = 0 is T-Com where is arbitrary. The general solution of X" +pX01 X ay+by where a andare a arbitrary, Below you will be asked to find Therefore we have w(x, 1) = (ayı + bye where we have incorporated into a and b. For this problem we consider the cases (p=0.02) - <0, and (p >0. (1) For p = 0 For p = - you wewe will containk anda, b) (x1) = wobei Forp bour answer will containk anda, b) (x1) = coobidos

Answer 1

Solution! We have already given that U(, t) = (að,+672)e Pt jis a general solution of the partial differential equation where y, and I are independent solution of the equation. ___ x"768 = 0 Now we consider three cases - 1 for p=o, we have the differential equation - xo Whose general solution is X = auto - ulat) = (antbe = Tantbl (2) for p =-k²o, we find from (1), X'_k²x=0 whose general solution is - X(x) = aelezetbe lize 2. ure,t) = [aeket bekx Jekt (3) for p = 4²>o, we find from (1), x" th ²x=0 whose general solution is X (21) = a cos(ke) th Sin (ke) :. U6u, t = [acos(kx) +bsin(use) Tékt