MAP 4305 Exam Practice 4

MAP 4305 Exam Practice 4 - r = sin r =-1 Are there any...

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Practice Exam 3: MAP 4305 * 1. Does 5 xy 00 + 4(1 - x 2 ) y 0 + y = 0 , x > 0 , have a solution which is bounded near zero? Notice that to answer this question, you only need to consider the indicial equation. 2. Determine the form of a series expansion about x = 0 of 2 linearly independent solutions to: x 2 y 00 - xy 0 + (1 - x 2 ) y = 0 , x > 0 . Do not find a recursion formula for the coeFcients. 3. Find the first three non-zero terms in a series expansion about x = 0 of 2 linearly independent solutions to: 3 xy 00 + (2 - x ) y 0 - y = 0 , x > 0 . 4. Draw solutions in the ( x,y ) plane of the following system in polar coordinates:
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Unformatted text preview: r = sin r =-1 Are there any non-trivial periodic solutions? If yes, are they limit cycles? If there are non-trivial periodic solutions, how many are there, and what can be said about their stability? 5. The Legendre polynomials P n ( x ) satisfy the following recurrence relation: ( n + 1) P n +1 ( x ) = (2 n + 1) xP n ( x )-nP n-1 ( x ) . Given that P ( x ) = 1 and P 1 ( x ) = x , determine P 2 ( x ), P 3 ( x ) and P 4 ( x ). * Instructor: Patrick De Leenheer....
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This note was uploaded on 04/07/2008 for the course MAP 4305 taught by Professor Deleenheer during the Spring '06 term at University of Florida.

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