exam2-solution

# exam2-solution - ENAE404 Spring 2006 Answers to Exam#2 75...

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April 27, 2006 75 minutes L. Healy 1. [20 points] For each of the following sets of diﬀerential equations with dynamical variables x , y , z , determine whether the equations divide into sets of equations that are decoupled from each other or if all three equations must be considered as one set, for each of the sets, whether the equations are linear or nonlinear, if there is a one-equation linear set, under what condition it is stable or unstable, if there is a stable one-equation linear set, what its frequency of oscillation is. (a) ˙ x + αyz =0 ˙ y + βxz =0 ˙ z + γxy =0 Nonlinear, doesn’t decouple (b) ˙ x + αz =0 ˙ y + βy =0 ˙ z + γx =0 Linear; y decouples from x - z . Motion in y is stable if β is imaginary, frequecy is | β | . (c) ˙ x + αx =0 ˙ y + βy 2 =0 ˙ z + γyz =0 Motion in x is linear and decoupled from others. Motion in y is nonlinear and decoupled from others. Motion in z is nonlinear and coupled to y . Motion in x is stable if real part of α is positive < ( α ) > 0, frequency is imaginary part = ( α ). 1

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exam2-solution - ENAE404 Spring 2006 Answers to Exam#2 75...

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