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11. Separation of Variables. By usingu(x, t) = X(x)T(t) or u(x,y, t) = X(x)Y(y)T(t), separate the following PDEs into two or three ODEs for X and T or X, Y, and T. The parameters c and k are constants. You do not need to solve the equations.
NOTE: one of the equations cannot be separated. Indicate this when you discover that equation.
(a) utt = (xux)x
(b) Utt = C2Uxx
(c) Ut = k(u + Uyy)
(e) Ut + cu ku
(f) Ut = k(yu + XU)
Separation of variables is demonstrated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.