Question

Please just explain parts d and f. Part d is 0 and f is true

16. (16 points) Suppose the displacement u(x, t) of a piece of flexible string is given by the initial- boundary value problem t> 0 100uzz = Utt, 0 <3 < 4, u(0,t) = 0, u(4,t) = 0, u(a,0) = 0, u(x,0) = 0(x) +0. (a) (2 points) Give a physical interpretation of Ox). (b) (3 points) In what specific form will the general solution appear? (1) ule,t) = Cn sin (5944) sin (176), (2) u(, 1) = C,cos (Barc) sin (172), (3) u(, t) = ŽCsin (5974) cos (478), (2) ulv,t) =Ž Ccos ( 5074 ) cos (15,6). n=1 n=1 n=1 (c) (4 points) Fill in the blanks: The coefficients of the solution in part (b) above can be found using the integral Cn = a (x)F(7x)dx. Jo a = - B=. y =- F(x) = cosine / sine (Circle the correct function.) (d) (3 points) Use the boundary conditions to determine the steady-state displacement v(x) of this string (e) (2 points) Does lim u(x, t) exist? What is it? 0 (f) (2 points) True or false: If ((x) = 0 the solution of the initial-boundary value problem will be the steady-state displacement for all t > 0.

Answer 1

function of ne Wheree 100 Unexe = Utt — 100 et les 100 meteen Trial solution is u = xT --- ® x is a - T is a function of t 24 = x.!! o X.5" — 24 = x't de XT — ☺ from egM® 4G, put is equo X-T'E 100x". T " The I" - K] conokrat) - Let Kaave value Go Karma CS Scanned with CamScanner

- m2 x = -mx x = 0,02 + 2 Si2x I" = -m2 the-m7.100 = c cos lomt te simtomt) Som is UCX,T)= X.T u = C, coome+Casinma) .( Gcoolomt. +4 sintomt) Applying boundary condition 1) 4(0,t)= 0 X=0,4= OF G C G Co>10mt+ (y sinsomt] Cico] 44(X.) - wen sin mo [ com 1omt + C4 Sin 20mt] - de-canne

22 (46=0 22 u(4,4)=0 x240 U=0, x=4 Os (2 sin 4m ſ 6 consomt +44 sintomt] sin 4m-0 Am ФЛ 1 Where na 1, 2, 3 ... 4 Turx.7) = C, sin nine) ( 300 40 ppt + Gysin lo mert 7 3) u(x,0) = 0 . *=0, u=0 :0 = Ca Sin Mono) [ +0] [C3=0) U(X,T) = cq sin paha). ( sin 10 (hat) - put C2 - Cq = 65 lucx.) = Cs. sin (na). sin (mat) CS Scanned with CamScanner where n=1,2,3---0

TIA 2) u(x,0)=0 t=0, U=0 Os C2 sinmäe [ 33+] sinma = 0 ma=nT Im=1 cs Scanned with CamScanner