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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.
- (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 that
is 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.
William W. Farr
Thu Oct 24 15:15:28 EDT 1996