1 a 2 X n 1 n 2 n 1 a n 2 x n X n 1 a n 1 x n so a 2 0 The recurrence relation

1 a 2 x n 1 n 2 n 1 a n 2 x n x n 1 a n 1 x n so a 2

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1 a 2 + X n =1 ( n + 2)( n + 1) a n +2 x n = X n =1 a n - 1 x n , so a 2 = 0 The recurrence relation satisfies ( n + 2)( n + 1) a n +2 = a n - 1 or a n +2 = a n - 1 ( n + 2)( n + 1) , so a 2 = a 5 = a 8 = ... = a 3 n +2 = 0 with n = 0 , 1 , ... For the sequence, a 0 , a 3 , a 6 , ... with n = 1 , 4 , ... a 3 = a 0 2 · 3 , a 6 = a 3 5 · 6 = a 0 2 · 3 · 5 · 6 , a 9 = a 6 8 · 9 = a 0 2 · 3 · 5 · 6 · 8 · 9 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (18/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Airy’s Equation 3 Airy’s Equation : The general formula is a 3 n = a 0 2 · 3 · 5 · 6 · · · (3 n - 1)(3 n ) , n 4 For the sequence, a 1 , a 4 , a 7 , ... with n = 2 , 5 , ... a 4 = a 1 3 · 4 , a 7 = a 4 6 · 7 = a 1 3 · 4 · 6 · 7 , a 10 = a 7 9 · 10 = a 1 3 · 4 · 6 · 7 · 9 · 10 The general formula is a 3 n +1 = a 1 3 · 4 · 6 · 7 · · · (3 n )(3 n + 1) , n 4 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (19/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Airy’s Equation 4 Airy’s Equation : The general solution is y ( x ) = a 0 1 + x 3 2 · 3 + x 6 2 · 3 · 5 · 6 + · · · + x 3 n 2 · 3 · 5 · 6 · · · (3 n - 1)(3 n ) + · · · + a 1 x + x 4 3 · 4 + x 7 3 · 4 · 6 · 7 + · · · + x 3 n +1 3 · 4 · 6 · 7 · · · (3 n )(3 n + 1) + · · · -10 -8 -6 -4 -2 0 2 -0.5 0 0.5 1 1.5 Ai ( x ) Bi ( x ) x Airy’s Equation Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (20/24)
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Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Chebyshev’s Equation 1 Chebyshev’s Equation is given by (1 - x 2 ) y 00 - xy 0 + α 2 y = 0 Let α = 4 and try a solution of the form y ( x ) = X n =0 a n x n , so y 0 ( x ) = X n =1 na n x n - 1 and y 00 ( x ) = X n =2 n ( n - 1) a n x n - 2 These are inserted into the Chebyshev Equation to give: (1 - x 2 ) X n =2 n ( n - 1) a n x n - 2 - x X n =1 na n x n - 1 + 16 X n =0 a n x n = 0 Note that the first two sums could start their index at n = 0 without changing anything Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (21/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Chebyshev’s Equation 2 Chebyshev’s Equation : The previous expression is easily changed by multiplying by x or x 2 and shifting the index to: X n =0 ( n + 2)( n + 1) a n +2 x n - X n =0 n ( n - 1) a n x n - X n =0 na n x n + 16 X n =0 a n x n = 0 Equivalently, X n =0 [( n + 2)( n + 1) a n +2 - ( n ( n - 1) + n - 16) a n ] x n = 0 or X n =0 ( n + 2)( n + 1) a n +2 - ( n 2 - 16 ) a n x n = 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (22/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Chebyshev’s Equation 3 Chebyshev’s Equation : The previous expression gives the recurrence relation : a n +2 = n 2 - 16 ( n + 2)( n + 1) a n for n = 0 , 1 , ..
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