Question

Problem 2. Find the type, transform to normal form, and find the solution u(x,t) of the ID wave equation, Utt = Uxx, with the initial conditions u(x,0) = 2sin 2x and ut(x,0) = 0 and the boundary conditions u(0,t) = u(nt,t) = 0.

Answer 1

du d+ d'u 1.C : u(x,0) = 2 sin22 u,(,0) = 0 ulo,t) = u(t) = 0 Ažu da² + Bau dray i care + Ddu & Edu = Fu opy Here, from eg. du 11 2x2 Scanned with 0-4(1)(-1) = + 4 CamScanner so ( PDE is Hyperbolic CS

-0 [urot) = 4(1,4) +0 Јвс Əx 4 (2.0) = 2 Sin 2x 2 te IC Wing Seperation of variables: (Product Method) u(a,t) = f(a) G (+) tu du fä & du FG at dx² putting in above obtained exprenions ی ای: 0 FS F - K G {k is arbitrary constan this gives "- kr =0 @ KG = 0 4 44. ☺ should satisfy B.C U(0,t) → FO) G (t) = 0 u(,) F(TU) G (t) = 0 Solving for non-trivial solution , ie Scanned path banga; F = A + Berk o G(+)#0 fonamen

taking ko - F" + p'FO F(x) A compr& Bsinpx Fo= A=0 & then f(n) = BainpIT zo Bto, . Sin p =0 pa na Sun T {n intragen? for sinna where an = na General Sahu an lt) = B ... Bisit then, 4...(0,4)=(.col.4 + 8* sin dut) sin mx (+) = Brconant + But gindat at t=0 u2,0) = 2 sin 24 Un(2,0)= 2 sin2x = Bn sinna at n=2 B = 2 0 = {Bintat in & in But content) sinne .. tro O = A.B. sinna So, u(x, t) = B.com) + sive pic ur,t) = 20 2t sin 2
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