415a1.1 - Kiam Heong Kwa 03/31/09 Kiam Heong Kwa 03/31/09...

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Kiam Heong Kwa 03/31/09
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Math 415A: Corrections to Section 1.1 Problem 1.1.11. It is NOT true that if y 0 = y (0) < 0 , then y ( t ) → -∞ as t → ∞ . This is not true since it can be shown that y ( t ) = 4 y 0 e 4 t 4 + y 0 ( e 4 t - 1) . Hence for y 0 = y (0) < 0 , y ( t ) is not defined at t = t * = 1 4 ln p 4 - y 0 - y 0 P . Hence it does not make sense to talk about the limit as t → ∞ . HOWEVER, it can be checked that lim t t * - y ( t ) = -∞ . In other words, y ( t ) approaches -∞ in finite time if y 0 < 0 . So, the only thing one can infer from the direction field of the equation y p = y (4 - y ) are the following statements: 1. If y (0) > 0 , then y ( t ) converges to 4 . 2. If y (0) = 0 , then y ( t ) = 0 for all t 0 and thus converges to 0 . 3. If
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415a1.1 - Kiam Heong Kwa 03/31/09 Kiam Heong Kwa 03/31/09...

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