final_spring_03

# final_spring_03 - Student Name Answers MA-366 Spring 2003...

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MA-366 Spring 2003, Final Exam Instructor: P. Stefanov May 6, 2003 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 15 2 15 3 20 4 24 5 21 6 12 7 18 8 10 9 15 TOTAL 150 1

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MA-366 Spring 2003, Final Exam 2 1. (15 pts) Find the general solution to (1+3 x ) y ±± - 3 y ± - 2 y =0 by means of formal power series about x 0 = 0 (do not study convergence/interval of convergence). Use the method of undetermined coeﬃcients: ±nd the recurrence equation, and ±nd the ±rst four non-zero terms in each of two linearly independent solutions. Answer: Recurrence equation: ( n + 2)( n +1) a n +2 +3( n + 1)( n - 1) a n +1 - 2 a n ,n , 1 , 2 ,.... Solution: y ( x )= a 0 ± 1+ x 2 + 1 6 x 4 - 1 5 x 5 + ... ² + a 1 ± x + 3 2 x 2 + 1 3 x 3 + 1 30 x 5 + ² .
MA-366 Spring 2003, Final Exam 3 2. (15 pts) Solve the IVP y ±± - 4 y ± y + y =0 , y (0) = 1 ,y ± (0) = - 1 , by means of formal power series about x 0 = 0 (do not study convergence/interval of convergence). Find only the ±rst four non-zero terms of the Taylor expansion.

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final_spring_03 - Student Name Answers MA-366 Spring 2003...

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