f08236ps2 - 92.236 Engineering Dierential Equations...

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92.236 Engineering Diferential Equations Practice Exam # 2 Fall 2008 Please write all answers and all work in the blue book provided. PLEASE SHOW ALL WORK! You will not receive ±ull credit i± you do not show your work. Problem 1. (25 points) Consider the autonomous d.e. dx dt = x 3 - x 2 - 6 x . a. Find all critical points (equilibrium solutions) of this d.e. x 3 - x 2 - 6 x = 0 x ( x 2 - x - 6) = 0 x ( x + 2)( x - 3) = 0 . The critical points are - 2 , 0 , and 3 4 pts. b. Draw the phase line (phase diagram) for this d.e. 10 pts. The three critical points divide the phase line into 4 intervals: x > 3 , 0 < x < 3 , - 2 < x < 0 , and x < - 2. dx dt v v v v v x =4 = 4(4 + 2)(4 - 3) > 0, so the direction arrow points up for x > 3. dx dt v v v v v x =1 = 1(1 + 2)(1 - 3) < 0, so the direction arrow points down for 0 < x < 3. dx dt v v v v v x = - 1 = - 1( - 1 + 2)( - 1 - 3) > 0, so the direction arrow points up for - 2 < x < 0. dx dt v v v v v x = - 3 = - 3( - 3 + 2)( - 3 - 3) < 0, so the arrow points down for x < - 3. -2 0 3 x c. Determine whether each critical point is stable or unstable.
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This note was uploaded on 11/01/2008 for the course MATH 236 taught by Professor White during the Fall '06 term at UMass Lowell.

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f08236ps2 - 92.236 Engineering Dierential Equations...

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