Exam 2 Solution

# Exam 2 Solution - Please e-mail typos and corrections to...

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Please e-mail typos and corrections to [email protected] PRELIM 2 , MATH 293 , SPRING 2006 STUDENT’S NAME: TA’S NAME: PROBLEM 1: PROBLEM 2: PROBLEM 3: PROBLEM 4: PROBLEM 5: TOTAL: 1

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2 PRELIM 2, MATH 293, SPRING 2006 Problem 0.1. Consider the differential equation dx dt = rx + x 3 containing the parameter r . ( a ) Calculate all the critical points for all the values of r and study their stability. ( b ) Find the bifurcation point. ( c ) Sketch the bifurcation diagram. Proof (a) Let f ( x ) = rx + x 3 = x ( x 2 + r ). The critical points are the real zeros of f . A critical point x is stable if and only if dx dt is increasing just to the left of x and decreasing just to the right of x , or, equivalently, f ( x ) < 0. (Why don’t they tell you this in the book?) Notice that f ( x ) = r + 3 x 2 . If r 0 then the only critical point is at x = 0, and it is unstable because f (0) = r 0. If r < 0, the critical points are at x = 0 , - r, - - r . The critical point at x = 0 is stable because f (0) = r < 0. The critical points at x = ± - r are unstable because f ( ± - r ) = - 2 r > 0. (b) The bifurcation point is the value of r at which the number of critical points changes. The bifurcation point is at r = 0. (c) The bifurcation diagram is a plot of the critical points against the parameter, namely
PRELIM 2, MATH 293, SPRING 2006 3 Problem 0.2. A jet ski is traveling across the water with an initial constant velocity of v 0 . At time t = 0 , the jet skier cranks the throttle, turning on the motor, which provides a constant acceleration of a 2 m/ s 2 , but the water resistance causes the jet ski to decelerate at ρ 2 v 2 m/ s 2 , where the velocity is v m/s.

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