Question

1. Solve the initial-boundary value problem one = 4 for () <<3, t> 0, u(0,t) = u(3, 1) = 0 for t> 0, u(x,0) = 3x – 2” for 0 < x < 3. (30 pts.)

Answer 1

44 = 4 Buxx Olacas 4101+) = U13,+) = o tipo u(x,0) = 3x - x2 osn23 Let u(a,t) = X(0) T (t) boo then from (1) me. - get, xt'=uX"T 54X"Trise ! .11. This XI s. . xlladx =0; T4ATEO I the coses am we considen following coses: 7) x(x2= A era ne Berana X101= AB20 X (3)= Alert - en 350] = & Aso => Bro which is trivial and so we cansiden next ease (iv) d 20 pollcon (= A 21+ B X10) I B=0 x(3) = 3 A 200 A 2a This is also trivial.

o Aco, darur. x (2) = e еаѕ o84р ма, хо) ге о xt 3) = 6 Ә) D Stn 3м го - да пт, . - х (х) = pain ти . - ( т'+ч м-т со теве – ук-. т () - Ке- чnn= Е. The genenal solution of a can be written as calxit = 3 Dn Sin nox e-unen it uto) = D він затя Date 1 Su-x2) Sin man die • 3 Го» с ва ба угода ) 23 J'a-s) екраз . 171 27 Sinnnn За – so п/ а ПИ

3 ..^7/3 mo/3 CE -3 sinna Sinnan n272 -X2 122 [-3.3mm + 2 } zimnega dn] en raz. Comment the most [i- eas no] 2 [1+cay wri] L na +(-unel - Dan-1 = 36x2 1 2n-1) 3 13 72 (2n-1) 3 17 3 ii. U (x, t) = 5 #2: e Sinatt 71 (20) - 4(21-17212 (2n-1)3113 Bin@11 7(20+) - Glo : May The general Solution of my is ularit te permis din myne e-bogotany 24 -4 (2n-1) Ulvet) = 1) Ha - 72. > hal (2n-1) 33