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Unformatted text preview: MATH 216 FINAL EXAM December 17, 2009 Please write your name: 4 points Section:  The test contains 8 problems worth 100 points total. To get the full credit you have to show your work. Page 12 contains a table of useful Laplace transforms. Problem Points Score 1 12 2 12 3 12 4 12 5 12 6 12 7 12 8 12 Total Typeset by AMS 'lEX 1 Name:  (12 points) 1. Problem. Find all critical points of the system of differential equations for functions x(t) and y(t) below and determine their stability x' = x 2 3y + 2 y' = xy x. X?.. 2 == 0 x ,) o (l) '>< .::; 0 I' (_1/ :::._ l/ lt 1 9/ (o, 3) " 0 2. j = .3 x==.1 ex _, _x)] /C I 1. 'A :=.. l =0 UAshb& L 2 o ( \) I l ' L: 3] [\/1): \ 2 Name:  (12 points) 2. Problem. By analyzing the type and stability of the critical point at (0, 0) or otherwise, match the systems of differential equations for functions x(t) and y(t) below with their direction fields on the next page. I 1) X yl x+y xy Figure D 3) xl 2x + y Figure jf J= y 1 x2y  5) x 1 = 2x + y F' E y 1 = x + 2 Y 1gure __ I) 2) I 2) X yl I 4) X yl I 6)x yl [2A \ \ ("+2.. 0 \ 2 fl. j '>+ 1.. = ':!: \ y) [ \ l '"L 1;.. ('A+\) t" l ;! 0 \ l L 5) l2 \')>. == 0 ..... LA (A I) + \  0 I 3 Figure ...1.... X L..t y x+y c Figure xy x+y Figure ___]2__ x+y A ("1) ____ ..._,....__....__...........
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