exam2 - force f t = cos(2 t Is there still a transient...

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Exam II — Math 220 — November 8, 2005 Name: Directions: Show all your work on these papers, and circle your answers if that’s appropri- ate. 1. A spring with a mass of 10 grams stretches a spring 4 cm. Using Hooke’s law, find the spring constant k . If there’s no damping or friction, what’s the natural frequency of this oscillator? 2. A damped oscillator is described by the differential equation x 00 + 3 x 0 + 4 x = f ( t ) . (a) If the driving force f ( t ) = cos(2 t ), find the general solution to this equation. (b) Identify (circle and label) the transient and steady-state solutions. (c) Suppose the damping constant decreases to 0, and the system is still subjected to the
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Unformatted text preview: force f ( t ) = cos(2 t ). Is there still a transient solution? Describe what happens. 1 3. Using the definition of Laplace transform, show that if L ( f ( t )) = F ( s ) , then L ( f ( t )) = sF ( s )-f (0) . 4. Solve the following ODE, using Laplace transforms: y 00 + 3 y + 3 y = u 1 ( t ) y (0) = 0 , y (0) = 0 2 5. (Extra credit) Compute the Laplace transform of a square wave - i.e., f ( t ) = +1 for t between 0 and 1, between 2 and 3, between 4 and 5, and so on, while f ( t ) =-1 for t between 1 and 2, 3 and 4, etc. (So f ( t ) is an infinite series.) Try to express F ( s ) in closed form. 3...
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