Assign9 - × 2 matrix and ~a is a constant vector b Find...

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AMath 250 ASSIGNMENT 9 Fall 2008 Note that this is not the last assignment. Submit all problems by Noon on Tuesday, November 25 th in the drop boxes across from MC4066. All solutions must be clearly stated and fully justified . From Problem Set 5, do: # 2 (Except (b)(i) which you did in the last asst.) # 4 (All parts; recall solution to (i) was c 1 e - t [1 1] + c 2 e - 5 t [1 - 1]) Also do the following Oscillator Problem: Suppose a mechanical oscillator obeys the IVP ± y 00 + 4 y 0 + 5 y = 0 y (0) = 0 , y 0 (0) = 1 (1) a) Write this as a first-order vector IVP, that is, an IVP of the form ± ~x 0 = A~x ~x (0) = ~a (2) where A is a constant 2
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Unformatted text preview: × 2 matrix and ~a is a constant vector. b) Find the solution to (2), hence write down the solution to (1). c) Sketch the phase orbits of the general solution to the vector DE in (2), highlighting the solution curve to the IVP (2) for t ≥ 0. d) Based on your phase portrait, is the motion under-, over-, or critically damped? e) Is the speed of the mass at a maximum just before, after, or as it crosses the equilibrium position for the first time? [Hint: speed = | y | ; use (c).]...
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This note was uploaded on 10/30/2011 for the course AMATH 250 taught by Professor Ducharme during the Fall '09 term at Waterloo.

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