Question

Now suppose f' is continuous and f" is piecewise continuous on (0, L). (b) If f(0) = f(L) = 0, then O f(x) = į bn sin ηπα L 0<x<L, n=1 612 Chapter 11 Boundary Value Problems and Fourier Expansions with bn = 2L n272 S“ r"(a)sin пах dr. L (11.3.5)

Answer 1

utt = 64 lax -① ocas3, t70 by separation of variades ugut) = XG). TE) Then ② implies = ff 711 XT 64x"T - = 0) = -8 @ where 64T separation Constant t= -p, pyo. X Aco is a 2 x"+p²x=0 and tht 64 P T zo by solving these ode's x=c, cos(px)+(₂ siu (px) and and T= Cz Cos&pt) + Cq Sin (8Pt) B.CS: Uit) 200 XC6)=0=611462.0 (6=0 and ubit)zo X(3) 20 = CZ sinBP) = siuln ii) Р: MIT 3 na112131- 1 + :: Xina Con sin (mm. a) and Tina chicos EquiT t Ja cü singarnart) (enicni ch are constants) Now cualt)= xn'in= An cos Ent 2) + Bn sine nyt t) ] sin heya) chocina An & Cnicu=Bn, by principle of superposition U(X16) - ГАм 4910)=0 >> nal12 "An cosent :) + Bn sin (mist AG nal 3 3 E nel + emma. I sincerity) Bnzo

. Olt) = ? An Cos(8ņrt). sin ma) - @ nzl and finally unio) = x29x = 8 An sin (max met ) which is fournier sine series of fea) = ²ar ,ocal. | Gaza (2390). sin Croatia) dx where Ana 2 3 cos SO = [Qande sesleniyo ); +6w) (cos (mga) ) 3 3 nh 20242 - (3729) Esiunton) - 6 sin nin) RO 34 n474 (: Sinozol h373 sinnazo . Goze cos ga). 3 n² 13 third 12.33 nu 3 conto (all terus will be zero except term at 223 123) 12-27 Ein n3 73 n3 73 oscanaut ). sininen 324 = En ď En " alt) = 324 E п3 n=1 n3