midterm1-A-solutions-public

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

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

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