Detrimine the equations-separation of variables

Question

Partial Differential Equations : Separation of Variables

Solve the following problems:

Separation of Variables. By using u(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_xx + U_yy)

(d) u_t=k(yu_x+u_y)

(e) U_t + cu_x= ku_xx

(f) U_t = k(yu_x + XU_y)