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Unformatted text preview: Differential Equations& Mathematica Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 Publisher: Math Everywhere, Inc. Version 6.0 DE.06 Systems and Flows LITERACY What you need to know when you're away from the machine. · L.1) Here's a rather detailed plot of a certain vector field:- 6- 4- 2 2 4 6 x- 4- 2 2 4 y Visible in this plot are four straight line trajectories and four other families of trajectories. Pencil in the straight line trajectories and one trajectory of each of the four other families. · L.2) You are given a system of differential equations x £ @ t D = 2- x @ t D 2 y £ @ t D = - y @ t D . You decide to look at the flow of the trajectories and you get:- 2- 1 1 2 x- 2- 1 1 2 y To get this plot, you plotted a vector field Field @ x, y D = 8 m @ x, y D , n @ x, y D< with tails at 8 x, y < for a selection of points 8 x, y < . The question here: What are the exact formulas for m @ x, y D and n @ x, y D ? · L.3) Here is the flow plot of a certain system x £ @ t D = m @ x @ t D , y @ t DD y £ @ t D = n @ x @ t D , y @ t DD together with a trajectory resulting from starter data x @ D = - 1.3 and y @ D = 1.7:- 2- 1 1 2 x- 2- 1 1 2 y Which way is the trajectory going? In what sense does the flow plot guide the trajectory along its way? · L.4) Here is the flow plot of a certain system x £ @ t D = m @ x @ t D , y @ t DD y £ @ t D = n @ x @ t D , y @ t DD of differential equations shown with two curves:- 3- 2- 1 1 2 3 x- 3- 2- 1 1 2 3 y One of these curves is a genuine trajectory. The other is bogus. Identify the trajectory and discuss the information that led to your choice. · L.5) Can two trajectories of a system x £ @ t D = m @ x @ t D , y @ t DD y £ @ t D = n @ x @ t D , y @ t DD ever cross each other like this? Why or why not? · L.6) The system of differential equations is x £ @ t D = x @ t D 2 + 3 y @ t D y £ @ t D = x @ t D- y @ t D 2 You are given that a trajectory 8 x @ t D , y @ t D< passes through 8 2, 1 < at a certain time t = t * . Your job is to write down the vector 8 x £ @ t * D , y £ @@ t * D< which is tangent to the trajectory at 8 2, 1 < . · L.7) a) Explain this: When you convert the single second order differential equation y ££ @ t D + 2 y £ @ t D + 8 y @ t D = into a system of two first order differential equations by putting x @ t D = y £ @ t D , you get x £ @ t D = - 2 x @ t D- 8 y @ t D y £ @ t D = x @ t D b) Are plots of solutions y ££ @ t D + 2 y £ @ t D + 8 y @ t D = the same as plots of trajectories of the system x £ @ t D = - 2 x @ t D- 8 y @ t D y £ @ t D = x @ t D ?...
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