Question: Please provide all steps. Thank you! (1 0 points) Let u be the solution to the initial boundary value problem for the Heat Equation, 6,u(t,x)

Please provide all steps. Thank you!

(1 0 points) Let u be the solution to the initial boundary value problem for the Heat Equation, 6,u(t,x) = 463.140, x), t e (0, on), x E (0, 1); with initial condition 3, x 6 [0,1), u(0, x) = f (X) = 1 o, x e [ 1]. and with boundary conditions 6xu(t,0) = 0, u(t, 1) = 0. Find the solution 14 using the expansion 00 u(t, x) = 2 c" vn(t) wn(x), "=1 with the normalization conditions 0,,(0) = 1, w\"(0) = 1. a. (3/1 0) Find the functions w,\" with index n 2 1. w" = E b. (3/1 0) Find the functions U", with index n 2 1. 0,. = Z c. (4/1 0) Find the coefficients c,\" with index n 2 1. C&quot

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