M257-316Notes_Lecture5

# M257-316Notes_Lecture5 - Lectures 4,5 Ordinary Points and...

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Lectures 4,5 Ordinary Points and Singular Points Lecture 5 4.3 The Airy equation: Consider the Airy equation y 0 = xy . x = 0 is an ordinary point. y = n =0 c n x n ,y 0 = n =1 c n nx n 1 0 = n =2 c n n ( n 1) x n 2 n =2 c n n ( n 1) x n 2 = n =0 c n x n +1 m +1= n 2 n = m +3 n =2 m = 1 c 2 2 x 0 + m =0 £ c m +3 ( m + 3)( m +2) c m ¤ x m +1 =0 c 2 c m +3 = c m ( m +3)( m +2) m , 1 ,... (4.12) (1) c 0 c 3 c 6 . c 3 = c 0 2 . 2 ,c 6 = c 3 6 . 5 = c 0 6 . 5 . 3 . 2 9 = c 0 9 . 8 . 6 . 5 . 3 . 2 c 3 n = c 0 (3 n )(3 n 1)(3 n 3)(3 n 4) ... 9 . 8 . 6 . 5 . 3 . 2 (4.13) y 0 ( x )=1+ x 3 3 . 2 + x 6 6 . 5 + ··· + x 3 n (3 n )(3 n 1) 3 . 2 + (2) c 1 c 4 c 7 . c 4 = c 1 4 . 3 c 7 = c 1 7 . 64 . 3 c 10 = c 1 (10 . 9)(7 . 6)(4 . 3) (4.14) c 3 n +1 = c 1 (3 n + 1)(3 n )(3 n 2)(3 n 3) (7 . 6)(4 . 3) y 1 ( x )= x + x 4 4 . 3 + x 7 7 . 6 . 4 . 3 + + x 3 n +1 (3 n + 1)(3 n ) 4 . 3 (4.15) y ( x c 0 y 0 ( x )+ c 1 y 1 ( x ) Radius of Convergence: lim m →∞ c m +3 c m | x | 3 =ln m →∞ | x | 3 ( m + 3)( m < 1 ρ = . (4.16) See BD for expansion of Airy Solution about x 0 =1 y ( x a n ( x 1) n .

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M257-316Notes_Lecture5 - Lectures 4,5 Ordinary Points and...

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