Question

2. Use the method of separation of variables to solve the boundary value problem ( au = karu 0<x<L t > 0 (0,t) = 0, > 0 (1.1) -0. > 0 (u(a,0) - (x) 0<x<L. Be sure to detail exactly how f(x) enters your solution E-

Answer 1

Ut = kUxq -90 Let une,t)= XC) TCH be Boletion of Then Ur=X114 Uxx=x' Bromo XT = "x"= =) $1 = -(804) => xl_dx=04T-TO Ugrot)=0 > XOTCH=0 ) x10=0 Ux (2,4)=0 >> x 7H30 =)X00 Cage (1) x-x=0 then x204tho The Batalions are word=924@4706 x')=a xoso =) Cho Woo Xa): 1. xecto Hence XQ) = c 18 constant fention urach is constant solution . Goerine-so sae de pe then X"_Bxsod 7 pktzo The solutions are x(c) = Ge PAQEP Toca pent 1. xlor=piePe poppen x(0)20 = pare-perro cogoo Ø x(2) 20 =) pcrepe peepboo = cele celo ®

@ t® hoks if 43046-0 1 X60)zo is trivial solution Cagecas? tco Sayda-ply then x'4px=0 & T4KPTOO The Babelion ale xG)= Gospetasimpe ___TOT= gene Xma-pasinpe+pc@spt X60)=0 =) -DCi sino tpe asoco => PUO> so x 220 >-Pasinpito-0 -)Simpl=o=Sinnte => pablo X00ecome tesinieso) Xn0)=COD (chose gol) Ton(t)= ane kiht (choose Gaan) Let un CC,+)= Xn6) ) fiet u tan uneet) Then pula,t)= an e A CERTY is solution of Got UCXO)=PBT) Hemm prxsen aneesmine They are an comment This is poulies Cosine series in (0,6) where an - ) Pare) cosmige dr.