Question

22: Solve the follwing boundary value problem Ugex - 2 = Utt: 0 < x < 1, t> 0, u(0, 1) = 0, u(1,t) = 0, 0 < x < 1, u(x,0) = x2 - x, ut(x,0) = 1, t > 0.

Answer 1

Ques @ Uxie - 8 = 4€t ; oc decl, t?o.. U10;t ) = ullit)=0,02x21 U(20) = x²x, 4+ (%,0)=1, to formula for : sol*: - Sineen Non-Homogeneous wave legma is -! n=1 ulkst) = Ž sin ng as Don eos e re. E + En sin (antiejt } + I sin (n)= * fnew) since-s )age] ds Lo Where Do = 1 912) din nqx dx En !ger) sin ni zdx 0 et facts a l's everyomega de .l=1, 0629= x2, 91x) = 4.4 = 1,(2343 byo Dm = 2 ( x-x) sim naz da sinnrz) dx -la sinnta dal x sinnri Lush ny71 / n2r2 N343 -21-x² cosnka, 2x/ ( Sna - CONX+Sinnrel ntcn22 = 2 [£1. losna 7 sinne ta empresas)-{-070 + y hon}]] ML + nt Sinnt near 2 + 2 losna 2 7013)

na sinnah n2t2 -ot sinna 77 "n221 l losna = (-1)? f simon = 0. L nr - 313 on = nene pa (4)?] 0 Now En- & ger, sinnet de [-(1)"} - ... + fact) = de l'flostysin ma di La | C-2) simnażde 12 of country'] = a[interesting

U not 2 4125) Ens t) - Sinaia using 0,048 in 890.0.4 : 1=1,4% acest) - sim poz. on en mart + En sinntz + Ž cinne se fa ley sin [c44) wr] ds Puente – Ž cimarep at 13 il, sest ta ei voca"} sinet'] + cimet '* * [6321] cimetom est) Ž [ca341) dimana pe cat desint] To t în aceste singure foam e-shpejds nel Ques@ Uxxt est=ut; oexca, tso Welle ulotte ucrtao' t20 U (220);!; 02x4!.. so!":- solution of Non-Homogeneous heat eqn is. G ginen by - l = j es eu tinhais-t) n²n2 kt s sinnax el fon (5)ds Lo e Sinnale le incids l J 1 uixit), = E Dr Simnas e le t n where.on = {j! que tinguts ļ & fricts = f* fent y aimanga.de

i l=x, K=1, 01x) = 1, #12t) = 2-3t.: us nxit -101 NT LI n ' La ) cosum =EJ? .. falta + 1 ca sin mp e da 11 inada n lo - an?t o e + sinna Anna mhes-t) ? folsyds com [1=643*) —o using eq" 0. £o in egno & also, (=, k=1 u(xt) = Ž On sinneemt = t (1-445"sinne en't + sinnix e st utsat) 2** (1-2-03") as - C-13]simnx [ 1 tayt pois os. nje ds I M8 CIS

-722 013 : + i] zues (non-humor 7=64000 6 се N=1 na -n X lole PIT res EN n=1 na n21 -n²t - X n=1 - 2.1+(3) smaz [ 2**set it as uerst) = , 2 cv") simex [ 07 $11-2*}] -727) 851--19"] na -72t - innal e te n2 ? n=1