Milestones_3_Math_33B_F08

Milestones_3_Math_33B_F08 - Mathematics 33B: Milestones for...

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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.

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Milestones_3_Math_33B_F08 - Mathematics 33B: Milestones for...

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