mid2-2010 - Question 1. [4 points] Consider the...

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Unformatted text preview: Question 1. [4 points] Consider the differential equation dx dt = x- a 2 x 3 . (a) Find all equilibria. (b) Determine the stability of each equilibrium point. (c) Draw the phase-line diagram for the differential equation. Solution. a) Equilibria occur when f ( x ) = x- a 2 x 3 = 0. Factoring, we have x (1- a 2 x 2 ) = 0 so x = 0 , 1 a . Version 1 : a = 2, so the answer is x = 0 , 1 2 . Version 2 : a = 3, so the answer is x = 0 , 1 3 . Version 3 : a = 6, so the answer is x = 0 , 1 6 . b) Differentiating, we have f = 1- 3 a 2 x 2 f (0) = 1 > f ( 1 a ) = 1- 3 a 2 1 a 2 < Thus, x = 0 is unstable, while x = 1 a are both stable. c) See Figure 1. 1 a- 1 a Figure 1: 1(c): x = 0 is unstable; the other two points are stable. Question 2. [6 points] Consider the equations Version 1 : 2 x 3- 8 x 2 + 11 x- 6 = 0 . Version 2 : 2 x 3- 10 x 2 + 15 x- 9 = 0 . Version 3 : 2 x 3- 12 x 2 + 19 x- 12 = 0 . 1 (a) Show that x 1 = 2 ( x 1 = 3) [ x 1 = 4] is a solution of the equation. (b) Use long division to find x 2 and x 3 , the other two solutions. (c) Calculate x 2 /x 3 , where w is the complex conjugate of w . (d) Find the four roots of x 4 = c 2 in the form a + ib , where a and b are real. (e) Express each root in part (d) in the form re i with r > 0. Solution. a) Version 1 : When x = 2, we have 2(2 3 )- 8(2 2 ) + 11(2)- 6 = 0. Version 2 : When x = 3, we have 2(3 3 )- 10(3 2 ) + 15(3)- 9 = 0. Version 3 : When x = 4, we have 2(4 3 )- 12(4 2 ) + 19(4)- 12 = 0. b) Long division factors the equation as ( x- a )(2 x 2- 4 x + 3) = 0 . Thus, using the quadratic formula, the other two solutions satisfy x = 1 p 16- 4(2)(3) 4 = 4 8 i 4 = 1 i 2 (You might notice some similarity between this and an assignment question!) c) It actually doesnt matter which solution is x 2 and which is x 3 , as the answer will be the same. If x 2 = 1- i 2 , then x 2 = 1 + i 2 and thus x 2 x 3 = 1 + i 2 1 + i 2 = 1 d) If x 4 = c 4 , then we have x 2 = c 2 or x 2 =- c 2 . From x 2 = c 2 , we have x = c . From x 2...
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This note was uploaded on 03/19/2011 for the course MAT 1332 taught by Professor Munteanu during the Spring '07 term at University of Ottawa.

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mid2-2010 - Question 1. [4 points] Consider the...

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