tt2-solns.pdf - APM346 TERM TEST 2 50 minutes Closed notes no external resources individual work Name Tutorial Student Number Exam policies \u2022 Closed

# tt2-solns.pdf - APM346 TERM TEST 2 50 minutes Closed notes...

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APM346 TERM TEST 2 50 minutes Closed notes, no external resources, individual work Name: Tutorial: Student Number: Exam policies : Closed book, closed notes, individual work. Indicate the main steps of your solution. This will increase you chances of getting at least partial credit. A bare numerical answer with no explanation may not receive full credit. Notation: C is the set of complex numbers, z denotes the complex conjugate of z . 1
2 APM346 TERM TEST 2 1. (20 points) Fix a constant c > 0, and consider the wave equation with Neumann boundary conditions ( 2 t - c 2 2 x ) u = 0 , 0 < x < 1 x u ( t, 0) = x u ( t, 1) = 0 , t R . (a) Determine all solutions of the form u ( t, x ) = T ( t ) X ( x ). (b) Compute the solution u to the above boundary value problem which also satisfies the initial condition u (0 , x ) = sin 2 ( πx ) , t u (0 , x ) = cos 2 ( πx ) , 0 x 1 . Solution. (a) T 00 T - c 2 X 00 X = 0 , so there exists λ R such that c 2 X 00 X = T 00 T = - λ. Get two ODE X 00 + λ c 2 X = 0 , X 0 (0) = X 0 ( L ) = 0 T 00 + λT = 0 .

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• Fall '08
• Chugunova

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