QUESTION 6 a) The propagation of a wave u(r, t) in a long cable is described...

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QUESTION 6 a) The propagation of a wave u(r, t) in a long cable is described by the following initial value problem Pu = 9 -00 <I<00, t>0 u(z, 0) = cos r, u(r, 0) = -0 <I< 00. Find the solution for this problem. b) A boundary value problem that represents the potential distribution u(r, y) over a rectangular plate is given by Fu Pu = 0, 0<r< n, 0<y<n ду? u(0, y) u(x,0) = 0, u(x, 7) = 2. = 0, u(T, y) = 0 %3D i. By assuming u(r, y) = X(x)Y(y), show that the partial differ- ential equation can be transformed into the following ordinary differential equations X" – AX = 0, Y" + AY = 0 where A is a constant. ii. Solve the above boundary value problem. QUESTION 6 a) The propagation of a wave u(r, t) in a long cable is described by the following initial value problem Pu = 9 -00 <I<00, t>0 u(z, 0) = cos r, u(r, 0) = -0 <I< 00. Find the solution for this problem. b) A boundary value problem that represents the potential distribution u(r, y) over a rectangular plate is given by Fu Pu = 0, 0<r< n, 0<y<n ду? u(0, y) u(x,0) = 0, u(x, 7) = 2. = 0, u(T, y) = 0 %3D i. By assuming u(r, y) = X(x)Y(y), show that the partial differ- ential equation can be transformed into the following ordinary differential equations X" – AX = 0, Y" + AY = 0 where A is a constant. ii. Solve the above boundary value problem.