Consider the steady state heat on a rectangle bounded by the lines x = 0, x...

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Consider the steady state heat on a rectangle bounded by the lines x = 0, x = 1, y = 0, and y = 1, given by J²u əx² J²u dy² = 0. Boundary conditions are given by: u(0, y) = 0, u(1, y) = 0, u(x, 0) = 0, u(x, 1) = (a) Show that the solution ∞ u(x, y) = ΣC₂ sin (näx) sinh (näy) n=1 X satisfies the boundary conditions u(0, y) = 0, u(1, y) = 0, and u(x, 0) = 0. [10 marks] (b) Find the formula for Cn, and calculate the first two coefficients. [15 marks] Consider the steady state heat on a rectangle bounded by the lines x = 0, x = 1, y = 0, and y = 1, given by J²u əx² J²u dy² = 0. Boundary conditions are given by: u(0, y) = 0, u(1, y) = 0, u(x, 0) = 0, u(x, 1) = (a) Show that the solution ∞ u(x, y) = ΣC₂ sin (näx) sinh (näy) n=1 X satisfies the boundary conditions u(0, y) = 0, u(1, y) = 0, and u(x, 0) = 0. [10 marks] (b) Find the formula for Cn, and calculate the first two coefficients. [15 marks]