Question

In Exercises 11-15, solve the nonhomogeneous wave initial-boundary-value problem. In each case, start by letting u(x,t) = T.(t) sin nz and proceed from there. n=1 11. u = Una + sin , u(,0) = sin 3.0, U (2,0) = sin 52, u(0,t) = u(Tt, t) = 0.

Answer 1

301,- given that stort by letting a (at) = Ž Tn (t) sin nal Now, (A Let Uax = Uux toina Flat) = sinx [(-1] data) t(a) = U14,0) = sin(3x) gla) = 4(x,0) = sin (59) ca ( given U10,t) = ULTIt) = 0 rit 7 By using a D'Abmest y Dulamal's primples together U14+) = { $(@t)+718-t) + a-t gla) ++++0] + [ a+CHY) F (Gr) dgdr x-1(A) Sin 55.ds Il sin 318++)+sin 318-+)/ + ť, x+(+) yat 3 + S sing d& dr by-lt-Y)

Now, we have to Calculate I, I, X Iz Separaterly firstly I :- I, = [ s in 3 (2+42 + sin 3(x+] x ( sin 3a) Cos(3+) sin (32) Cos(at) I, att In = sinl55) s. a-t = coss (att) + cos (x-t) I - (2 sin (52) - sin (5+)) txt(ter) I, = sta Sin (ag.de -tt-Y) t $ (-COS (E)) 2++->) x-(tr) + 11 cose xt (+-p) )-COS (2-(tv))] dr 1 * [coses -Linsin us t 11 2 sin (2). sin (+-r] do

t 2 sin (n) sin Lt-r) do 1 -zsin (1) 4 asin n cost - 25in 1a) + 2 sintal cos(t). - wh 3 Subtitute Above I , Iz XI 3 In Equ I= sin (30) Cos 13t) - Sin (52) Sin (54) – Sin la) 75ini cosct) 1117,t) = sin 132). Cosl3t). sinl52) sin (5+) -sin la) tuinea) Coslu)