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Unformatted text preview: Mathematics 33B: Milestones for Week 3 2.9, 3.1, 3.3 Note: We started by finishing up our discussion on Existence and Unique ness. This took up most of the lecture...you should know how to do problem 25 and 29 of 2.7 that we did in class. Topic 1: 2.9 autonomous equations and stability. This section is quite straightforward. A. An autonomous differential equation is one that has the form: y = f ( y ). B. So long as f is a continuous function we can expect solutions and so long as f is differentiable we can expect that those solutions are unique. C. What is the structure of the solution set? 1. Translates of solutions are solutions: y = f ( y ) is solved by z = z ( x ). Then let w ( x ) = z ( x + s ). We have that w ( x ) = z ( x + s ) = f ( z ( x + s )) = f ( w ) so a translate of a solution is a solution (and hence all solutions are translates of solutions). 2. For each y , if f ( y ) = 0 then y = y is a solution to the initial value problem y = f ( y ) and y ( x ) = y . This is called an equilibrium solution for....
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This note was uploaded on 02/08/2010 for the course MATH 33B taught by Professor Staff during the Winter '07 term at UCLA.
 Winter '07
 staff
 Math

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