Question

Differential Equations

6. Solve the following boundary value problem:

?? = 3???, 0 < ? < 1, ? > 0; ?(0,?) = ?(1,?) = 0; ?(?, 0) = 7 sin ?? − (1/9) sin 3?x

Answer 1

09100 ounce > U(+) = Ullt): 0 Ulm.u) = fuxo Bounduzy Condition -Initial condition. where t is tine nie tishlacement, unt) is tembeauture of the system crot. ) We solve by the method of separation of variable U101.+25 H12). 66+) (take nun trivial deln ) = Hico).6(+) Ut act? 460 Unn Hlin.6(H) tic onlar, cct) F"(07 и(х) ic.lt Uro.t)= 0 = H(o).60(+20 > H(o)=0 ullot)=0 » HID).Gita - Ulla H" (0-dn- H"(to n col-AntB 16olo B= nisl - An HOITAX = A- cer, Ar. 외 wint 1207 ip teivial sol

Calenco 2 to H-WH = m²-u2 = o m 2+M ucala cenntçenia Hol - ct ucll ciene + cell Ceno cella cile EMO) 0 . C=0 (2= trivial feln H = U13 ct)=0 o cii2 dco ecoin Hiltun - metu? = m=& Mi H C, Cosnatc. sinun (+ (= C₂ sinul Sinu Sinul = 0 Sinyl E shina The friviulsus =104 NT hot nt I-dok ni N = dn ملاح

Hr Cn Singleton 1-IN = 64) Kolt) is (+)-dku(t)= Gilti= Aenkt -ntr- .K+ IL X Uncaut) = Hn (42. holt Un cont) = Dr. Sinnan na kt е 22 NAN the principal of supe a position Uco.tz Unnit) Unit): On Sinnan -h2172 14 n=1 n=1 VOD.02 E on 2in1.51 fin! - Dor Singh noi I Dn - from Sinne ndon

# 3UXX o excl UCU. D = V1 +) = 0 U(2,0) = 7 Sinha- Kliquis k=3. here l=1 +(x)=4&inux-1 sinana then. WxH) = sin ne se va = { on sinnra e hill? (31) 11 whose 3 D = +1x). Simning x 6 Di ja finna linarx) singud Dn zal f(y sinnix-t sinane ). signnast Let tp мах, Jx td

), 7. Sint-I sin3t ). sinni dt S 7 D. || 유기 Sint sinnt dt - 아 Sin3t.8inni ddt 야 0 1) 11 [ Sinma. sinna 구 it mtn C 름 출명을 cind Dn = o. VnEINI (13) thin Nxt = 7 sinna è 37² j Sin3m.11 어 Ans! | UAt) A sinni.E37 아 in 37