Question

Let u be the solution to the initial boundary value problem for the Heat Equation, дли(t, х) — 5 дғи(t, х), te (0, co) хE (0, 1); with initial condition хе х, u(0, х) %—D f(x) 1 хе 2 and with boundary conditions u(t, 0) 0 дди(t, 1) 3 0. Find the solution u using the expansion u(t, х) = "(х)"n ()"а ", n=1 with the normalization conditions | Un(0) 1 Wn = ]. (2n - 1). a. (3/10) Find the functionss wn, with index n > 1. sin(2n-1) pi/2x) b. (3/10) Find the functions v/n, with index n > 1. M exp(-5((2n-1) pi/2)^2t) c. (4/10) Find the coefficients cn, with index n > 1. Сп — W

I want just c^n

Answer 1

Heat uct(x)ニ) Cn stnican-r)4 We hane -) a)=ΣCn sinl (21-t x serhes fouater sine By よ2) sin(2n-1) L 2 2 Mert 1sin (an-)ォxdx Sn 2 l Sin (2n-1) 7 d

+Sin(en-A 71 2n-1 (2n1)A09 22 gin (ana (Osc2n-t) p 2 2n-1 COS(2n1 Tcos(2n-t 2 (2n-1a/ 2/4 (2n-122 Sin (2n-) Cos(2n-1) f2 2n-) Cn Sin (2n1)212 C2n 71 Cos (2n-1) A