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# f08236ps2 - 92.236 Engineering Dierential Equations...

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92.236 Engineering Differential 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 full credit if 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 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x =4 = 4(4 + 2)(4 - 3) > 0, so the direction arrow points up for x > 3. dx dt vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x =1 = 1(1 + 2)(1 - 3) < 0, so the direction arrow points down for 0 < x < 3. dx dt vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = - 1 = - 1( - 1 + 2)( - 1 - 3) > 0, so the direction arrow points up for - 2 < x < 0. dx dt vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle x = - 3 = - 3( - 3 + 2)( - 3 - 3) < 0, so the arrow points down for x < - 3.

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f08236ps2 - 92.236 Engineering Dierential Equations...

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