415aq3 - Solution to Quiz 3, Math 415A 1. Find solution to...

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Unformatted text preview: Solution to Quiz 3, Math 415A 1. Find solution to the wave equation utt = uxx for 0 < x < π satisfying initial and boundary conditions: u(x, 0) = sin x + 1 sin(2x) , 7 ut (x, 0) = 0 , u(0, t) = 0 = u(π, t) Solution: Separation of variable gives un (x, t) = Xn (x)Tn (t) with Xn (x) = sin(nx), Tn (t) = cn cos(nt) + kn sin(nt). Since ut (x, 0) = 0, kn = 0. un (x, t) = cn sin(nx) cos(nt) 1 Noting only two modes are present corresponding to c1 = 1, c2 = 7 , we obtain u(x, t) = sin x cos t + 1 sin(2x) cos(2t) 7 2. Transform the following equation into a system of first order ODEs: y ′′ + 4y ′ + 4y = e−2t (t + 1) Solution: Define x1 = y , x2 = y ′ . Then the system is: x′ = x2 , x′ = −4x2 − 4x1 + e−2t (t + 1) 1 2 3. Verify that x = 1t e is a solution to 1 d x= dt Solution: We note that LHS = RHS = So, verification complete. 2 3 −1 −2 2 3 −1 x −2 et et = LHS d1t e= dt 1 et et = 2 et − et 3 et − 2 et 1 ...
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