midterm1_solutions_08 - Ordinary Differential Equations...

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Ordinary Differential Equations Math 22B-002, Spring 2008 Midterm 1: Solutions NAME ............................................................ SIGNATURE .................................................. I.D. NUMBER ............................................... No books, notes, or calculators. Unless stated otherwise, show all your work. Question Points Score 1 10 2 10 3 20 4 20 5 20 6 20 Total 100 1
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1. [10%] By use of the general theorems given in class, say what you can about the existence, uniqueness, and t -interval of definition of a solution y ( t ) of the ODE (ln | t | ) y 0 + e t y = t 2 - 9 t - 4 with the initial conditions: (a) y (2) = 5; (b) y (5) = 2. (Don’t try to solve the ODE explicitly.) Solution The equation is linear, of the form y 0 + p ( t ) y = g ( t ) , with coefficient functions p ( t ) = e t ln | t | , g ( t ) = t 2 - 9 ( t - 4) ln | t | . The function p ( t ) is discontinuous at t = 0, where ln | t | is discontinuous and at t = ± 1, where ln | t | = 0. The function g ( t ) is discontinuous at t = 0 , ± 1 and at t = 4, where t - 4 = 0. (a) By the general existence-uniqueness theorem for linear IVPs, a unique solution y ( t ) exists and is defined in the interval 1 < t < 4, which is the largest t -interval containing the initial time t 0 = 2 where the coefficient functions are continuous. (b) Similarly, a unique solution y ( t ) exists and is defined in the interval 4 < t < , which is the largest t -interval containing t 0 = 5 where the coefficient functions are continuous.
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