Differential Equations Lecture Work Solutions 142

Differential Equations Lecture Work Solutions 142 - u(0 t =...

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5.3 Forced Vibrations Problems 1. Consider a vibrating string with time dependent forcing u tt c 2 u xx = S ( x, t ) , 0 <x<L subject to the initial conditions u ( x, 0) = f ( x ) , u t ( x, 0) = 0 , and the boundary conditions u (0 ,t )= u ( L, t )=0 . a. Solve the initial value problem. b. Solve the initial value problem if S ( x, t )=cos ωt . For what values of ω does resonance occur? 2. Consider the following damped wave equation u tt c 2 u xx + βu t =cos ωt, 0 <x<π, subject to the initial conditions u ( x, 0) = f ( x ) , u t ( x, 0) = 0 , and the boundary conditions
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Unformatted text preview: u (0 , t ) = u ( π, t ) = 0 . Solve the problem if β is small (0 < β < 2 c ). 3. Solve the following u tt − c 2 u xx = S ( x, t ) , < x < L subject to the initial conditions u ( x, 0) = f ( x ) , u t ( x, 0) = 0 , and each of the following boundary conditions a. u (0 , t ) = A ( t ) u ( L, t ) = B ( t ) b. u (0 , t ) = 0 u x ( L, t ) = 0 c. u x (0 , t ) = A ( t ) u ( L, t ) = 0 . 142...
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