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M251Hex2_mockup(sp09)

M251Hex2_mockup(sp09) - b 9 ≤ t< 3 t 2 t ≥ 3(b Find...

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Math 251H Second Midterm Exam 75 minutes March 23, 2009 SAMPLE EXAM. 1
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1. A 3 kg mass is suspended from a spring, which stretches the spring 5 m from its natural length. The system is placed into the liquid with damping constant 14 Newton-seconds per meter. At t = 0, the system is at rest at its equilibrium position, then receives an external force of 6cos( ωt ) Netwons. Assume ω is positive and g = 10 m/s 2 . (a) Set up an initial value problem that describes the motion of the mass. Be sure to explain any variables that appear in your equation. (b) For which value of ω will the system have resonance? (c) Find the steady-state solution.
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2. (a) Find the Laplace transform of f ( t ) = braceleftBigg
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Unformatted text preview: b 9 , ≤ t < 3 , t 2 , t ≥ 3 . (b) Find the Laplace transform of f ( t ) = e t sin( √ 2 t ) . (c) Find the inverse Laplace transform of F ( s ) = s-2 s 2 + 2 s + 10 . 3. Solve the following initial value problem using Laplace transform, y (4)-16 y = 0 , y (0) = 1 , y ′ (0) = 2 , y ′′ (0) = 0 , y (3) (0) = 0 . 4. Solve the following initial value problem, y ′′ + 6 y ′ + 13 y = g ( t ) , where g ( t ) = b t-1 , ≤ t < 1 , , t ≥ 1 . sketch the solution in time. 5. Solve the following initial value problem. y ′′-2 y = 4 e − 2 t δ ( t-1) , y (0) = 0 , y ′ (0) = 1 . Sketch the solution....
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