final_22b_07 - Ordinary Differential Equations Math 22B-002...

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Ordinary Differential Equations Math 22B-002, Spring 2007 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 ): y 0 + 2 ty = e - t 2 , y (0) = y 0 . (b) For what initial-value y 0 is y (2) = 0? 2
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2. [20 pts.] (a) Solve the initial value problem yy 0 + 1 = t, y (6) = 3 . (b) For what t -interval is the solution defined? 3
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3. [20 pts.] (a) Find the equilibrium solutions of the equation y 0 = y ( y - 2) 3 . (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) = - 1 behave as t + ? How does the solution with y (0) = 1 behave as t → -∞ ? 4
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4. [20 pts.] (a) Find the general solution of the equation y 00 - 4 y 0 + 3 y = 0 . (b) Find the general solution of the equation 2 y 00 + 2 y 0 + 5 y = 0 . 5
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5. [20 pts.] Suppose that ω 0 , ω , F 0 are nonzero constants, and consider the ordinary differential equation y 00 + ω 2 0 y = F 0 cos( ωt ) .
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