final_22b_sample_07

final_22b_sample_07 - Ordinary Differential Equations Math...

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Ordinary Differential Equations Math 22B-002, Spring 2007 Sample Final Exam NAME .................................................................... SIGNATURE .......................................................... I.D. NUMBER ....................................................... No books, notes, or calculators. Show all your work Question Points Score 1 20 2 20 3 20 4 20 5 20 6 20 7 20 8 20 9 20 10 20 Total 200 1
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1. [20 pts.] (a) Solve the following initial value problem for y ( t ) in t > 0: y - 2 t y = t, y (1) = 2 . (b) How does y ( t ) behave as t 0 + ? 2
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2. [20 pts.] (a) Solve the initial value problem y + (cos t ) y 2 = 0 , y (0) = y 0 , where y 0 is an arbitrary constant. (b) For what values of the initial data y 0 is your solution defined for all -∞ < t < + ? 3
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3. [20 pts.] Consider the ordinary differential equation y = ( y - 3) ( y 2 - 1 ) (a) Sketch a graph of the right-hand side of this equation as a function of y , and find all equilibrium solutions of the equation. (b) Sketch the phase line of the equation, and determine the stability of the equilibria you found in (a). (c) How does the solution with y (0) = 0 behave as t + ? How does the solution with y (0) = 2 behave as t → -∞ ?
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