Quiz-5-Math-20D-W09solutions

Quiz-5-Math-20D-W09solutions - Quiz #5 Math 20D March 6 Two...

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Quiz #5 Math 20D March 6 Two problems ( § 5.4, #5 and § 5.6, #3). SOLUTIONS. 1. ( § 5.4, #5. 10 points) Find all singular points of the given equation and determine whether each one is regular or irregular: ( 1 - x 2 ) 2 y 00 + x (1 - x ) y 0 + (1 + x ) y = 0 . Solution : We rewrite this as (since ( 1 - x 2 ) 2 = (1 - x ) 2 (1 + x ) 2 ) 0 = y 00 + x (1 - x ) (1 - x 2 ) 2 y 0 + 1 + x (1 - x 2 ) 2 y = y 00 + x (1 - x )(1 + x ) 2 y 0 + 1 (1 - x ) 2 (1 + x ) y. The singular points are x = - 1 and x = 1 . (You may write p ( x ) = x (1 - x )(1+ x ) 2 and q ( x ) = 1 (1 - x ) 2 (1+ x ) . ) (a) For x = - 1 we have ( x + 1) p ( x ) = ( x + 1) x (1 - x )(1 + x ) 2 = x (1 - x )(1 + x ) is not analytic near x = - 1 . (We do not need to compute ( x + 1) 2 q ( x ) . ) So x = - 1 is an irregular singular point. (b) For x = 1 we have ( x - 1) p ( x ) = ( x - 1) x (1 - x )(1 + x ) 2 = - x (1 + x ) 2 is analytic near x = 1 . And we have ( x - 1) 2 q ( x ) = ( x - 1) 2 1 (1 - x ) 2 (1 + x ) = 1 1 + x which is also analytic near x = 1
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This note was uploaded on 08/07/2009 for the course MATH 20D taught by Professor Mohanty during the Spring '06 term at UCSD.

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Quiz-5-Math-20D-W09solutions - Quiz #5 Math 20D March 6 Two...

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