midterm1_solutions_08

midterm1_solutions_08 - Ordinary Differential Equations...

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Unformatted text preview: 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 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 + 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 + 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 = 2 where the coefficient functions are continuous....
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This note was uploaded on 03/05/2009 for the course MATH 22B taught by Professor Hunter during the Spring '08 term at UC Davis.

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midterm1_solutions_08 - Ordinary Differential Equations...

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