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hw2 - Consider the first-order equation dx dt = x 2-1(1(a...

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Math 454 Problem Set #2 Fall 2008 Sect. 1 Due Thursday, 9/11, at the beginning of class. Reminder. You are encouraged to work together on homework problems. How- ever, the final write-up must be your own. You will be quizzed on some of these problems, so be sure you can do all the problems without help. Also, unless otherwise stated, all reading assignments are from the text. Read: Sects. 2.5-7; 3.0, 3.1 Problems (note that the problems for Ch. 2 begin on p. 36 of the text): - Sect. 2.5 #1, 4, 5(b), 6 - Sect. 2.6 #1 - Sect. 2.7 #1, 4 - Sect. 3.1 #1, 3 P1. (Finding the blow-up time)
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Unformatted text preview: Consider the first-order equation dx dt = x 2-1 . (1) (a) Show that Z x 1 x dx x 2-1 = Z t 1 t dt (2) holds for all trajectories x ( t ) with x ( t ) = x and x ( t 1 ) = x 1 . (b) Suppose a trajectory starts at x = 2 at t = 0 . At what time does it reach x = ∞ ? (c) For a general initial value problem ˙ x = f ( x ) , x (0) = x , suppose f ( x ) > for all x ≥ x . Can you formulate a condition for x ( t ) → ∞ in finite time? Hint: You can state your answer in terms of an integral similar to that in Eq. (2)....
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