Question

(1 point) Solve the nonhomogeneous heat problem Ut = uzz + 4 sin(5x), 0< I<T, u(0, t) = 0, u(T, t) = 0 u(x,0) = sin(3.c) u(x, t) = Steady State Solution lim, , u(x, t)

Answer 1

not Rogst IG ucot) - uceito of Non-homo heat equation ₃ 4-xuan Dcx, t),o cual, tzo you, o) - fire) then solution is ut) & dim(bax sol. of Heather (t-1) falds е where, Engl fons sinfining the Fn ♡ (277) are CS Sin (pix) eranzet na + 9 sinfo it xn22 e 2 Scanned with CamScanner

SOMS Page No. Date give - Wext As insa, or clorul والمعاملہول Asin sa ucalpel Singx Compaiting with with formula, we get altxt)= 7.Sinsas. e=n fowa Singre Now En foxes. Sin (none) dve À L Sin3ze. Sin2) des . Sinu sinna dre 풍 Mote fasi Sinax Sinbreche if n=a otheowise So, bom eq@ ef n=3, then Ez = 2/4 sinzu.sinande = 212 = 1 ie. 텅 . Nowo In = 2 & Cre, it) sin(nr , che and En Per nt3 Fi Sin52) S(ng-lx & t Es =* l'"s Sinse sinna dre Using eq82 of n=5, Es = 4 sinsa Sin Suda & (2) Skanned with CamScanner Is = 4 and in=0 for n#5

solution is uch, t = 24 En sin(none) et penet na 48 Sin (nog) en (1214) Shay Frlds using Es= and Enzo for n&3 Ę = 4 and E fernts Sol 7.(132712 Jt 4card) 5 Sin(3.12} 2013 (75) + Sin (som many Eslaids com a (Sing).e-9t + Sinsx -256+-3) falds Š -Sin 3 ze -94 + Sin Six.67225) 255 do e ds Scanned with Sama seantet

Sinux + 4e2st Dale uca, A)= eat Sin 3xt -25t 4 e Sin Sn [ eerst 25 Jo प = e Sinsa ast E 25 e atsinau k = e + 4. Sin Sze [l-e-25+) which is required solution Lim ucre, & tto - Lim [ent Singte + 4 Sinette t-100 Since Lima osin3x) + 4 finstellno) 25 too so tin ura= too Lim 466) = Z Scanned with CamScanner A sin 536 25 cs
The given differential equation is non homogeneous heat equation of 1 dimension. The formula for the solution given in attached file. With the help of that formula we find solution.