M294F09WS_Last

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online