final_spring04

# final_spring04 - Student Name Answers MA-366 Spring 2004...

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MA-366, Spring 2004, Final Exam Student Name: Answers Note: You may use scratch paper. All solutions and answers must be on those sheets in the space provided. If the space provided is not enough, continue on the back or attach additional sheets (well organized). You must show your work. Two sheets of handwritten notes are allowed. No calculators. Problem # Max Score 1 12 2 12 3 12 4 18 5 12 6 27 7 12 8 12 9 12 10 12 TOTAL 141 1

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MA-366, Spring 2004, Final Exam 2 Problem 1. (12 pts) Solve the system x 0 D ± ± 3 ± 1 ± 1 ± 3 ² x , x (0) D ± 4 1 ² . Plot the solution in the x 1 x 2 -plane. Show the directions along the trajectoriy. What is the type of the zero solution? Is it stable? Answer: x D 3 2 e ± 2t ± 1 ± 1 ² C 5 2 e ± 4t ± 1 1 ² , the zero is a stable node.
MA-366, Spring 2004, Final Exam 3 Problem 2. (12 pts) Find the general solution of the system: x 0 D 0 @ 30 0 5 ± 20 00 3 1 A x What can you say about the stability of the zero solution? Answer: x D c 1 e ± 2t 0 @ 0 1 0 1 A C c 2 e 3t 0 @ 1 1 0 1 A C c 3 e 3t 0 @ 0 0 1 1 A . The zero solution is unstable because some of the eigenvalues are positive.

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MA-366, Spring 2004, Final Exam 4 Problem 3. (12 pts) Find the general solution of the system: x 0 D ± 11 ± 41 ² x Sketch the phase portrait of the system. Show clearly the directions. What is the type of the zero solution? Is it stable?
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final_spring04 - Student Name Answers MA-366 Spring 2004...

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