Use the separation of variables, u(x, y, t) = X(r)Y(y)T(t), to solve the following vibrating mem- brane problem Ju Ət² J

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Use the separation of variables, u(x, y, t) = X(r)Y(y)T(t), to solve the following vibrating mem- brane problem Ju Ət² Ju J²u + = ər² əy² in a domain with 0<x< 1 and 0 <y < 1. The boundary conditions are u(0, y) = 0. u(1, y) = 0 u(x,0) = 0, u₂(x, 1) = 0 and the initial conditions are f(x,y) u(x, y,0) u₁(x, y, t) = 0