**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.

- (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.
- (10 pts) Suppose that is an
*n*by*n*real matrix. Suppose that is an eigenvector of corresponding to a real eigenvalue . Show thatis a solution of the linear system

- (15 pts) Let . Solve the IVP with
*x*(0) = 1 and determine the interval of existence. - (15 pts) If is the matrix
compute the first five terms in the series for the matrix .

- (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? - (15 pts) Let . For the matrix
find the general solution. Then find the solution for the initial condition

- (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.

Thu Oct 24 15:15:28 EDT 1996