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Unformatted text preview: Instructor: Wayne Hacker Math 550.391-f08 Homework Set 2: ODEs on the real line Difficulty Rating Scale: Next to each problem will be the following labels. • E = Elementary Level Problem • I = Intermediate Level Problem • D = Difficult Level Problem Definition: An ODE of the form ˙ x = f ( x ), where ˙ x = dx dt is an autonomous equation. Recall: A differential equation of the form ˙ x = f ( x ) with initial condition x (0) = x can be thought of as a vector field and the solution x = x ( t ) as the trajectory of a particular fluid particle that ”flows along the x-axis” and is labelled by ”tagging the fluid particle” at time t = 0 at the point x . Flows on the line: For problems 1-5 our goal is to use a geometric approach whereby we interpret the differential equation of the form ˙ x = f ( x ) as the graph of the function v = f ( x ), where v = ˙ x plays the role of the value of the velocity of the trajectory x = x ( t ) (i.e., the slope of the solution). We then plot the graph of the function in the xv-plane and use the sign and size of v to plot the trajectory of the particle, denoted by arrows, along the the x-axis. That is, we superimpose the path of the particle over the x-axis. Math 550.391, Homework Set 2-f08, Hackernotes c Wayne Hacker 2008. All rights reserved. 2 Instructions: For problems 1-5 complete each step listed below (as part a,b, and c). BeFor problems 1-5 complete each step listed below (as part a,b, and c)....
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- Fall '08
- Math, Malthusian growth model, Logistic function, population growth model