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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 tex2html_wrap_inline63 is an n by n real matrix. Suppose that tex2html_wrap_inline69 is an eigenvector of tex2html_wrap_inline63 corresponding to a real eigenvalue tex2html_wrap_inline73 . Show that

    displaymath53

    is a solution of the linear system

    displaymath54

  3. (15 pts) Let tex2html_wrap_inline75 . Solve the IVP with x(0) = 1 and determine the interval of existence.

  4. (15 pts) If tex2html_wrap_inline63 is the matrix

    displaymath55

    compute the first five terms in the series for the matrix tex2html_wrap_inline81 .

  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 tex2html_wrap_inline87 . For what initial conditions does the theorem guarantee a unique solution to this equation?

  6. (15 pts) Let tex2html_wrap_inline89 . For the matrix

    displaymath56

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

    displaymath57

  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.

    tex2html_wrap95





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