Question

5. Find a solution u(x, t) of the following problem utt 0 u(0, t) — и(2, t) — 0 2 sin 3T и(а, 0) — 0, и (х, 0) — sin Tz _

Answer 1

Gkven tt phw,h guen predeu i Cltt uhere o S2 boundary aud fal The given Conditons as ulat) ut)-0 0)0, e(,0) = ShIT-2S3 Euation a Ene-dimnsienal Spation On. wave Vashables, lat uct)- x Tt) be the Slution t Gaton D hore xC)ortiont culy dTt)fnlion tt enly

e Na from uaton,ue have (Coataut Contistet Slutiont 9alioD tathg kp uaten he have IN K

ニ0 where DE dot fhea'r differesid Gaffet Guaion wth Contaat Aendliary Guetion ta giveu by otipti cohere o , B-p Complinustiny fenckion Cf) elosBxGSpr) cAlso,the gight -haid side Guatienis 3ero, therefda Paubicular tegra P.-O

Heute メ= Cf.+PI メ=GspxtGSithpa→6 Azdin fm , we have aT 2pT t (Dap ahere D's dt difeveucthalyton wth aa Catfhcdeuet gteu bmtp where d co, Bp Complinuy function, 1T

EGasptpt) CF lso hesight laud idet qationg herfre pasrtiudar fileg heule T CP.+PI T- Gos(pt Bpltto or TGlos pt)Ga(ipt) € Now fronn, aud, ue have toys b-(A*)n :* u(ot)- 4otoa}

fom (to), ehave unt)Gnp(pt)+G;f&©¢U® Sfapo Go 78hap SihT, n l,2, 3, --. E' 1uLu de n=1,2,3, - - have SroSatien we m,2,3, 2 Pottirg t-o ih Equali eneaue loto)CaShlo :ur)o n=12,3,. u(7)GS 0こ

2 'snoof m 2,3-- C3 O ation we haue Puty G ult) GC S ( S m2,3, - na12,3 Jaw hypiniplespeyportien, mat genal oluhion is jienby uit)- bnSh ( si 3 SkYY 2 at SiATL-2Sh3Tx Bn Sih Si)as3 whore Bn= bn and oso) BSY Sintua) tos(277)

Bs snua)Bm¥) 2 +& sth (3)hlH. +0Sf CuT o Sn ( shcua)tosh( +osh() )25( +osh ( Euoting the Corespordg tesue, howe B-, Be-2 aud B0, B, B B0 A4 we b , T ie. b bEfT)z ie, b aud e. 31 o, b30, b0,b0, b0, g

haue we utt) bshSE+&,5l) S/2Tt sit1 31Tt 2- (t - ult)ot () sk t0+0 -0t t04 s /a 3T totot 37T heule utrt) )ShCTE) a sduthonfu jiven onu difmesieu wave atienO