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MA 3431 A-96 Exam 1 Name: .

Show your work in the space provided. Unsupported answers may not receive full credit. If you use Maple to answer a question, make sure you include your Maple output with your exam and that you clearly identify which problem you are working on.

1. (15 pts) Explain the terms saddle, stable spiral, and stable node in the context of two dimensional linear systems of differential equations. Make sure you include information on eigenvalues and a sketch of a representative phase plane, including trajectories.

2. (10 pts) Suppose that is an n by n real matrix. Suppose that is an eigenvector of corresponding to a real eigenvalue . Show that

is a solution of the linear system

3. (15 pts) Let . Solve the IVP with x(0) = 1 and determine the interval of existence.

4. (15 pts) If is the matrix

compute the first five terms in the series for the matrix .

5. (20 pts) Describe our basic existence and uniqueness result for a single differential equation of the form x' = f(x,t). Include the conditions on f must satisfy to guarantee that the solution exists and is unique. Then consider . For what initial conditions does the theorem guarantee a unique solution to this equation?

6. (15 pts) Let . For the matrix

find the general solution. Then find the solution for the initial condition

7. (10 pts) On the direction field given below, sketch trajectories starting at the initial conditions (1,-1) and (-2,-1). Also, classify the origin as a sink, source or center.

William W. Farr
Thu Oct 24 15:15:28 EDT 1996