midterm1-B-solutions-public

midterm1-B-solutions-public - Math 201 Fall 2011 Midterm...

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Math 201 Fall 2011 Midterm Exam 1 (B) October 4, 2011 1. [4 marks] Does the differential equation ( x - 1) dy dx = 2 y 3 - 1 , y ( x 0 ) = 1 necessarily have a unique solution in some small interval ( x 0 - h,x 0 + h ) for the given values of x 0 ? Why? (a) x 0 = 1 ; (b) x 0 = 0 . The existence and uniqueness theorem has two conditions to check: f ( x,y ) = (2 y 3 - 1) / ( x - 1) is continuous, and ∂f/∂y = 6 y 2 / ( x - 1) is continuus for (a), near (1 , 1) , neither f nor ∂f/∂y is conitnuous at (0 , 1) , the theorem cannot be applied. So it does not necessarily have a unique solution for (b), both f and ∂f/∂y are continuous. So teh theorem applies, and there is a unique solution 2. [5 marks] Sketch the phase portrait of the autonomous equation dN dt = 1 - N 2 and classify each equilibrium point as stable, unstable, or semistable. Equilibria: 1 - N 2 = 0 , N = ± 1 These equilibria cut the phase space (the real line) into three intervals ( -∞ , - 1) , ( - 1 , 1) , (1 , ) In each interval
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midterm1-B-solutions-public - Math 201 Fall 2011 Midterm...

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