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9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0) = 0, U(TT, t) = 0, t> 0 $ (3,0)
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The given one dimensional wave: equation is the given boundary and initial conditions que 410,0)=0- 2) 4111,) = 0 -12) , wher3 XT — (8) from equations (1), 17) and (8), we have XT = CPXT CPT Let - I = k ( constant) 4 9) Since wave motion is a be(3) Equation (11) is a liveat differential equation with constant coefficients. Hence the auolilliasy equation of (11) is g(4) Again from lii) and (ili) of equatroy (10), we have till be et = -6 2, T1=-cobalt T + cpt = 0 a dI + ceply =o - ate to (we have cofo = (G Cogcbtt fi cp) Also the right hand side of equation (13) 13 Zelo, therefore particular solution The O.. % C414, t) = G-gruba)( 6 Cosebe) + G9ycht) _ 40 ciblelying the condition 4078, ) = 0, we have 41 Tyt)= G Silber) f G coskept) +Let G G4 = bn s... o from equation (18), we have. 418) = bin Sinha) Bifcht) hala, e the most general solution of equation (1)1 1 0 equating the collesponding terms, we have to Bi=1, Ba =0, B3 =0,--- .: We have Bi=1, Ba = B3=--- = b Cli) = 1, bz = 63=

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9. Solve the wave equation subject to the boundary and initial conditions u(0,t) = 0, u(x,0)...
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