ρ Two series can be added or subtracted for x x ρ f x g x X n 0 a n b n x x n

# Ρ two series can be added or subtracted for x x ρ f

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ρ > 0 Two series can be added or subtracted for | x - x 0 | < ρ f ( x ) + g ( x ) = X n =0 ( a n + b n )( x - x 0 ) n Products can be done formally for | x - x 0 | < ρ : f ( x ) g ( x ) = " X n =0 a n ( x - x 0 ) n # " X n =0 b n ( x - x 0 ) n # = X n =0 c n ( x - x 0 ) n , where c n = a 0 b n + a 1 b n - 1 + ... + a n b 0 Quotients are more complex, but can be handled similarly Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (13/24) Introduction Series Solutions of Differential Equations Example Review Power Series Properties of Series 2 Suppose f ( x ) satisfies the series below converging for | x - x 0 | < ρ with ρ > 0 f ( x ) = X n =0 a n ( x - x 0 ) n The function f is continuous and has derivatives of all orders: f 0 ( x ) = X n =1 na n ( x - x 0 ) n - 1 , f 00 ( x ) = X n =2 n ( n - 1) a n ( x - x 0 ) n - 2 , converging for | x - x 0 | < ρ The value of a n is a n = f ( n ) ( x 0 ) n ! , the coefficients for the Taylor series for f . f ( x ) is called analytic . If X n =0 a n ( x - x 0 ) n = X n =0 b n ( x - x 0 ) n , then a n = b n for all n . If f ( x ) = 0, then a n = 0 for all n Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (14/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Series Solution near an Ordinary Point Series Solution near an Ordinary Point, x 0 P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0 , where P , Q , and R are polynomials Assume y = φ ( x ) is a solution with a Taylor series y = φ ( x ) = X n =0 a n ( x - x 0 ) n with convergence for | x - x 0 | < ρ Initial conditions: It is easy to see that y ( x 0 ) = a 0 and y 0 ( x 0 ) = a 1 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (15/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Series Solution near an Ordinary Point Theorem If x 0 is an ordinary point of the differential equation: P ( x ) y 00 + Q ( x ) y 0 + R ( x ) y = 0 , that is, if p = Q/P and q = R/P are analytic at x 0 , then the general solution of the DE is y ( x ) = X n =0 a n ( x - x 0 ) n = a 0 y 1 + a 1 y 2 , where a 0 and a 1 are arbitrary, and y 1 and y 2 are two power series solutions that are analytic at x 0 . The solutions y 1 and y 2 form a fundamental set . Further, the radius of convergence for each of the series solutions y 1 and y 2 is at least as large as the minimum of the radii of convergence of the series for p and q . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (16/24)

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Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Airy’s Equation 1 Airy’s Equation arises in optics, quantum mechanics, electromagnetics, and radiative transfer: y 00 - xy = 0 Assume a power series solution of the form y ( x ) = X n =0 a n x n From before, y 00 ( x ) = X n =2 n ( n - 1) a n x n - 2 = X n =0 ( n + 2)( n + 1) a n +2 x n , which is substituted into the Airy’s equation X n =0 ( n + 2)( n + 1) a n +2 x n = x X n =0 a n x n = X n =0 a n x n +1 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Power Series Ordinary Point — (17/24) Introduction Series Solutions of Differential Equations Airy’s Equation Chebyshev’s Equation Airy’s Equation 2 Airy’s Equation : The series can be written 2 ·
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