test2_spring_04

test2_spring_04 - Name Answers MA-366 Spring 2004 Test 2...

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MA-366, Spring 2004, Test 2 Name: Answers Note: You may use scratch paper. All solutions and answers must be on those sheets in the space provided. If the space provided is not enough, continue on the back or attach additional sheets (well organized). Show your work. Calculators are not allowed. Problem 1. (12 points) Solve the differential equations: (a) 3y 00 C 2y 0 C y D 0 Answer: y D e ± t = 3 ± C 1 cos p 2 3 t C C 2 sin p 2 3 t ² (b) y 00 ± y 0 ± 2y D 0 Answer: y D C 1 e ± t C C 2 e 2t . (c) 25y 00 C 10y 0 C y D 0 , y . 0 / D 5 , y 0 . 0 / D 4 . Answer: y D 5e ± t = 5 . 1 C t / . 1

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MA-366, Spring 2004, Test 2 2 Problem 2. (15 points) A spring-mass system is governed by the equation y 00 C S y 0 C 2y D 0 ; where S ± 0 is the damping constant. (a) Find the least value of S , for which the mass will cross the equilibrium position at most once, regardless of the initial conditions. Explain. Answer: The characteristic equation is N 2 C SN C 2 D 0 . It has complex roots, if 0 ² S< 2 p 2 ; and non-positive real roots (repeated or not) for S ± 2 p 2 . In the ﬁrst case, the mass crosses the equilibrium
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test2_spring_04 - Name Answers MA-366 Spring 2004 Test 2...

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