ode219week3_1 - Math 219 Lecture 5 Ali Devin Sezer Contents...

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Math 219, Lecture 5 Ali Devin Sezer October 12, 2010 Contents 1 Autonomous equations and population dynamics 1 2 Picard Iteration 2 2.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The integral operator . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Autonomous equations and population dynam- ics Today we will first study the following concepts: 1. Autonomous differential equations, 2. Equilibrium points of an autonomous differential equation, 3. Stability analysis of equilibrium points. A differential equation of the form y = f ( y ) (1) is called autonomous. The word “autonomous” means having the right or power of self-government (see http://m-w.com ). The function y satisfying (1) is “autonomous” be- cause the time derivative dy/dt ( t ) of y , i.e., the velocity of y at any time t , is determined directly by y ( t ) itself; i.e., y determines its own velocity. 1
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The logistic equation dy dt = r ( y - y/K ) y. is an example of an autonomous equation. y = sin t + y, is not autonomous because the right side depends on t as well as y . One can graphically analyze an autonomous ode as follows: draw the graph of f ( y ) as a function of y and identify the zeros (those y for which f ( y ) = 0) of the function f . The zeros of f are called critical points or equilibrium points . If y 0
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