Separation of variables

Question

Assignment:

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) u_tt = (xu_x)_x
(b) U_tt = C²U_xx
(c) U_t = k(u + U_yy)
(d) u_t=k(y_u+u_y)
(e) U_t + cu_x = ku_xx
(f) U_t = k(yu_x + XU_y)