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Unformatted text preview: Consider the rst-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 nite time? Hint: You can state your answer in terms of an integral similar to that in Eq. (2)....
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This note was uploaded on 02/17/2010 for the course MATH 142532 taught by Professor Glasner during the Spring '10 term at University of Arizona- Tucson.
- Spring '10