As before a and a 1 are arbitrary with y 0 a and y 0 a 1 It follows that a 2 16

# As before a and a 1 are arbitrary with y 0 a and y 0

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As before, a 0 and a 1 are arbitrary with y (0) = a 0 and y 0 (0) = a 1 It follows that a 2 = - 16 2 a 0 = - 8 a 0 , a 4 = 4 - 16 4 · 3 a 2 = 8 a 0 , a 6 = 0 = a 8 = ... = a 2 n and a 3 = - 15 3 · 2 a 1 = - 5 2 a 1 , a 5 = - 7 5 · 4 a 3 = 7 8 a 1 , a 7 = 9 7 · 6 a 5 = 3 16 a 1 , ... Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (23/24)

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Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Chebyshev’s Equation 4 Chebyshev’s Equation with α = 4 : From the recurrence relation , we see that the even series terminates after x 4 , leaving a 4 th order polynomial solution. The general solution becomes: y ( x ) = a 0 ( 1 - 8 x 2 + 8 x 4 ) + a 1 x - 5 2 x 3 + 7 8 x 5 + 3 16 x 7 + ... y ( x ) = a 0 ( 1 - 8 x 2 + 8 x 4 ) + a 1 x + X n =1 [(2 n - 1) 2 - 16][(2 n - 3) 2 - 16] · · · · · (3 2 - 16)(1 - 16) (2 n + 1)! x 2 n +1 ! More generally, it is not hard to see that for any α an integer, the Chebyshev’s Equation results in one solution being a polynomial of order α (only odd or even terms). The other solution is an infinite series. The polynomial solution converges for all x , while the infinite series solution converges for | x | < 1. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (24/24)
• Fall '08
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