Question

Problem 3: Solve the following initial value / Neumann problem by separation of variables: (8 points) Unt -90.x = 0, (t, x) € Rx (0,2), u(0,x) = cos? ), (0, 3) = [1 – cos (3)], 1,(t,0) = 0,,(t, 2) = 0.

I need help

Answer 1

32 5 att - 9Uxx = 0,0 (t, x) E R X [0, 2] U (0,1)= 6,? () -e (0,2) = [1-G (9_))' -6 4,(4,0) = 0 Uxlt, 2) = 0 Sol2 let U (9,t)= XOTIH) XT" & Uxx = but the value of u in given & DE 2 we get 4 X"T 2) Utt T"X - qe TX"=0 T"X= 9TX" 2) -- =) T" 9T X" Х L.HS L.His in th of t only and Roll-s in f of only, so RHS if they are constant. 27 = 1 = (contant i.e * I" - and X 7 = 1 9T

& T" -a&T =0 X"-dx = 0 By egh (A) & (6) a (t,0)= U, (t, 2) = 0 tt = x'(0)T (4) = x'(2) TLH =0 X'10) = 0 & X'@) 20 (if T(=0 then uso BUP Solving X"_x=0 X'00) = 0 and X'(2)=0 Case - dzo x" & T!! =0 키 Xas= AxtB ten & TCL)= (Ę +D & X'(2) = A X'(X) A x'60) = A=0 X'(2) = A=0 So X(X) = B U = B (Ek+D) = A 2 + Bob 2 Ao = BC , BD=60 2 clet i=112 Case - 2 d>o , 41 X" 6-e²x = 0 4X » X(X) = Gelatge X'(X) = Gueux - cane X'lo) GM-C₂14 zo X' (2)

3 내 - more on "][9] =60 merry - we ney this has hon - trivial roll iff det of the matrix is near hoe om ve-20/= 0 -w2e-204 + 4² e zu -zill e te zu which is never posible . ( exponential fn) we don't have non-trivial noth qn this can. Cax-3 d=-412 co X" + m²y = 0 7 X(X) = G Concur +G Sinux X'(x) = -Gul Sinux + G4 Gaux als - G.0 + Gel sa =-Gu Sinzent Gun GA 2450 Hva x'co) X'(2) > G=0 ш 4 A24. -cu sinau Guru/3)=88) this System of egn has non-trivial rol" iff In a corona l=0 2) ursin zon 20

Sinach zo Crichto ey ) 2all =hT ly hl 2 NEW ㅋ nEN in= -** -- (hu)? Xhila) = Go (nome) bring the above eigin value in, we can nolue T equs T" -9&T in T"+9 () PT Ano Con (397 4 + DSin (3974 Sin (3017) New =) Ty(t) = An Cn3nt TI Now, combining the noth of Cau-s & Care -3 U (2,t) = Ao + Bot + E f., Go (9.00€ ) + bn Singhnt)] ne Gr() Nowy ucast) = Go? () - It Con(1) 2 So, Ź tz Ga(193) A. An CG (HTZ) ha - On (my)= mo + AI GA (TEX) + P2 Go (1 2) + P3 G(BIT)

Comparing can (6) & (7) we get Bo = 3 , 37 B = -2 Ghlt B = Ź B₂ = BA O Buzz z > > Bo = 3, B, = - 4 Ao al, Az = 1/2 3 IT By ㅗ 6nt A, = Az = ---=0 Alno Hence, u(xt) = I 2 + 3t t 2 Jo. Go (395) + 6 ) Sin (3.91 Jordan Go (GMP) + shop Sin (Can €) J. Ga (202) +0+- + for 1 + 3t Q z) u(xt): 5 st Sin (32) GA (72) + { Golfmk.) 69 (62) + Sin.Gnx) 6n