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 yx ax = = . Then 1 1 '( ) n n n n = = and 2 2 ''( ) ( 1) n n n nn a x = =− . Putting these into '' ' 0 yy −− = , 21 0 2 01 0 20 2 1 (1 ) 0 (2 ) ) 0 2[ ( 2 ) ( 1 ) ] 0 nn n n n n n n n n n x nax a x n a x a x aa n n a n x ∞∞ == = + = + = −= ++ = −+ + + − − = ∑∑ Comparing coefficients, (i.e. 0 2 2 a a = ) and for 1 n , 2 2 2 ) ) 0 ) ) ) 0 0 ) 2 n n n an a a a n a a n + + + = += =+ + So 2 2 n n a a n + = + is the recurrence relation. For even n , 00 2 4 4 6 44 28 6 642 4 8 a a a a = × = × × For odd n , 1 3 3 11 5 5 7 3 55 3 1 5 77 5 3 1 0 5 a a a a a a = = × = ×× Generally, for 1 k , 22 24 0 2 0 0 ( 2 2 ) 2 ( 2 2 ) 2 ( 1 ) 2 ( 1 ) 2! kk k k a a k k k a a k = = = ⋅− = " " "
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2 21 23 1 1 1 2 1 (2 1)(2 1) 1)(2 3 (2 )(2 2) 2 1)(2 )(2 1)(2 2) 3 2 1 2! 1)! kk k k aa a a k k k a a k k −− + == = = ++ + =⋅ +− + " " " " Hence 22 1 1 00 1 0 1 01 11 1 () ( 2 1 ) ! 12 ! ( ) ! k k k k yx a x a x a a ax a x k x k a x ∞∞ + + + + =+ ⎛⎞ + + ⎜⎟ + ⎝⎠ + ∑∑ The two linearly independent solutions are 2 0 1 k k k x k = and 0 1)! k k k k x k + = + , in general form. For 2 0 1 k k k x k = , the first four terms are 246 1 28 4 8 xxx + +++ " . For 0 1)! k k k k x k + = + , the first four terms are 35 7 3 15 105 xx x x + + " . 2. (a) Let 0 n n n = = . Then 1 1 '( ) n n n n = = and 2 2 ''( ) ( n n n nn a x = =− . Putting these into '' 2 ' 0 yy λ −+ = , 0 2 0 20 2 1 (1 ) 2 0 ) ) 2 0 2[ ( 2 ) ( 1 ) 2 ] 0 nn n n n n n n n n n x nax a x n a x a x n n a n x λλ = + = + = = + = + + + = Comparing coefficients, += (i.e. 2 ) and for 1 n , 2 2 ) ) 2 0 ) ) ) n n n a n a a na a + + + + For even n , 42 0 64 0 ( 4 ) 43 4 ! 24 ) ( 8 ) 65 6 ! a a −× = × ×
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3 For odd n , 31 1 53 1 75 1 21 ( 2 ) 32 3 ! 23 ( 6 ) ) 54 5 ! 2 5 ( 2)( 6)( 10) 76 7 ! aa a a a λ λλ −× =− × = × × Hence 22 1 1 00 24 6 0 35 7 1 () (4 ) ) (8 ) 1 2! 4! 6!
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This note was uploaded on 01/28/2011 for the course MATH 150 taught by Professor T.qian during the Spring '09 term at HKUST.

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solHW3A - MATH150 Introduction to Ordinary Differential...

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