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286practicefinal

# 286practicefinal - 286-E1 Spring 2010 Practice Final Exam 1...

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286-E1, Spring 2010 Practice Final Exam 1 1. (a) (10 pts) Find all a R so that the solution of the initial value problem x = x 4 - 1 , x (0) = a satisfies lim t →∞ x ( t ) = - 1. (b) (10 pts) Classify the type of origin as a critical point of the system: d dt x = 0 - k k 0 x where k = 0 is an arbitrary nonzero real number. Describe geometrically the trajectories of this system. 2. (15 pts) Find the general solution of the equation: ( x 2 + 1) dy dx + xy = x 3. (20 pts) Find the general solution of the equation: (2 x sin y cos y ) y = 4 x 2 + sin 2 y 4. (10 pts) Compute the Fourier series of the 4 π -periodic extension of the function: f ( t ) = 0 , 0 t 2 π 2 , - 2 π t < 0 5. Suppose that a mass m = 2 kg is attached to a spring with spring constant k = 32 N/m whose other end is fixed to a wall. (a) (10 pts) Find the value of damping coefficient c for which this system is critically damped. What is the general solution for the position function x ( t ) of the mass at time t in this case?

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286practicefinal - 286-E1 Spring 2010 Practice Final Exam 1...

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