solHW3A - MATH150 Introduction to Ordinary Differential...

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1 MATH150 Introduction to Ordinary Differential Equations Homework 3 Suggested Solution 1. Let 0 ( ) n n n y x a x = = . Then 1 1 '( ) n n n y x na x = = and 2 2 ''( ) ( 1) n n n y x n n a x = = . Putting these into '' ' 0 y xy y = , 2 1 2 1 0 2 0 1 0 2 0 2 1 ( 1) 0 ( 2)( 1) 0 2 [( 2)( 1) ] 0 n n n n n n n n n n n n n n n n n n n n n n n n n a x x na x a x n n a x na x a x a a n n a na a x = = = + = = = + = = + + = + + + = Comparing coefficients, 2 0 2 0 a a = (i.e. 0 2 2 a a = ) and for 1 n , 2 2 2 ( 2)( 1) 0 ( 2)( 1) ( 1) 0 ( 1 0) 2 n n n n n n n n n a na a n n a n a a a n n + + + + + = + + + = = + + So 2 2 n n a a n + = + is the recurrence relation. For even n , 0 0 2 4 0 0 4 6 4 4 2 8 6 6 4 2 48 a a a a a a a a = = = × = = = × × For odd n , 1 3 3 1 1 5 5 1 1 7 3 5 5 3 15 7 7 5 3 105 a a a a a a a a a a = = = = × = = = × × Generally, for 1 k , 2 2 2 4 0 2 0 0 2 2 (2 2) 2 (2 2) 2 2 2( 1) 2(1) 2 ! k k k k a a a a k k k k k a k k a k = = = = = = " " "
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