s251ex2(su06)

s251ex2(su06) - MATH 251 Summer 2006 Exam II July 20, 2006...

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MATH 251 Summer 2006 Exam II July 20, 2006 ANSWERS: 1. (a) T = π (seconds); (b) γ = 8 ; (c) Underdamped; (d) True. 2. D 3. C 4. B 5. D 6. (a) Y = ( A 0 t 3 + A 1 t 2 + A 2 t ) e t + A 4 e 3 t cos 2 t + A 5 e 3 t sin 2 t + ( A 6 t + A 7 ) cos 3 t + ( A 8 t + A 9 ) sin 3 t ; (b) y ( t ) = C 1 e t + C 2 e 3 t + t 2 + 8 3 t + 23 9 . 7. (a) The required initial value problem is: 2 u ′′ + 4 u + 10 u = 0 , u (0) = 4 , u (0) = 0 ; its solution is u ( t ) = 4 e t cos 2 t + 2 e t sin 2 t . (b) 2 ( rad/sec ) (c) lim t →∞ u ( t ) = 0 8. f ( t ) = ( u 2 ( t ) - u 4 ( t ))(3 t - 6) + u 4 ( t )(4 e 3 t ) = u 2 ( t )(3 t - 6) + u 4 ( t )(4 e 3 t - 3 t + 6) ; its Laplace
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