s251ex2(sp04)

s251ex2(sp04) - Math 251 Spring 2004 Midterm Exam II 1 a...

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Math 251 Spring 2004 Midterm Exam II 1. a) False; b) True; c) False; d) False 2. a) ω 0 = 4 (rad/sec) b) The beat frequency is ± ± ± ± ω - ω 0 2 ± ± ± ± = 4 . 5 - 4 2 = 0 . 25 (rad/sec). 3. y ( t ) = 1 2 e - 3 t - 1 2 e t + 3 4 te t 4. a) m > 1 2 (kg) b) The initial value problem is u 00 + 2 u 0 + 2 u = 0 , u (0) = 1 10 , u 0 (0) = 0 . Its solution is u ( t ) = 1 10 e - t cos t + 1 10 e - t sin t . c) μ = 1 (rad/sec) d) It’s overdamped. 5. y ( t ) = - 1 15 e 2 t + 1 40 e - 3 t + 1 24 e 5 t + u 1 ( t )( 1 4 e 5( t - 1) - 1 4 e - 3( t - 1) ) 6. a) f ( t ) = 1 - u 1 ( t ) b) 1 4 u 00 + 16 u = 1 - u 1 ( t ) , u (0) = 0 , u 0 (0) = 0 . c) u ( t ) = 1 16 - 1 16 cos 8 t - u 1 ( t )( 1 16 - 1 16 cos 8( t - 1)) 7. L{ f ( t ) } = 2 s 2 + e - 2 s ( 2 s 3 + 2 s 2
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