M294F09WS_Last

M294F09WS_Last - Math 2940 Worksheet 7.4 7.5 8.1 8.3 9.1...

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Math 2940 Worksheet: 7.4, 7.5, 8.1, 8.3, 9.1, 9.3 December 3, 2009 1. Let A be an n × n matrix and k a scalar. Consider the following two systems: d~x dt = A~x d~ c dt = kA~ c Show that if ~x ( t ) is a solution to the first system, then ~ c ( t ) = ~x ( kt ) is a solution of the second system. 2. Consider the interaction of two species of animals in a habitat. We are told that the change of the populations x ( t ) and y ( t ) can be modeled by the equations dx dt = 1 . 4 x - 1 . 2 y dy dt = 0 . 8 x - 1 . 4 y where time t is measured in years. (a) Sketch a phase portrait for this system. From the nature of the problem, we are interested only in the first quadrant. (b) What will happen in the long term? Does the outcome depend on the initial populations? If so, how? 3. Consider the dynamical system d~x dt = ± - 1 - 2 2 - 1 ² ~x (a) Find all real solutions of this system. (b) Solve the sytem with initial condition ~x (0) = ± 1 - 1 ² . 4. Find all the eigenvalues and ”eigenvectors” of the linear transformations and determine if the
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This note was uploaded on 12/25/2009 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell.

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M294F09WS_Last - Math 2940 Worksheet 7.4 7.5 8.1 8.3 9.1...

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