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Chapter 2

# Chapter 2 - II Series Solutions of Ordinary Differential...

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II Series Solutions of Ordinary Differential Equations Contents 1 Taylor Series 1 2 Analytic functions 2 3 Singular point and ordinary point 2 4 Cauchy-Euler Equations 4 5 The Frobenius Method 6 6 Bessel’s Equation 7 We have fully investigated solving second order linear differential equations with constant coeﬃcients: Ay ′′ + By + Cy = 0 , where A,B,C are constants. Now we will explore how to find solutions to second order linear differential equations whose coeﬃcients are not necessarily constant: P ( x ) y ′′ + Q ( x ) y + R ( x ) y = 0 . 1 Taylor Series Definition 1 The Taylor series about x 0 of a function f ( x ) is the series n =0 f ( n ) ( x 0 ) n ! ( x x 0 ) n . There exists R 0 such that the series is convergent in | x x 0 | < R and divergent in | x x 0 | > R . The number R is called Radius of Convergence . We have f ( x ) = n =0 f ( n ) ( x 0 ) n ! ( x x 0 ) n , | x x 0 | < R. Theorem 1 (Theorem 3 in the book) R = lim n →∞ a n a n +1 , a n = f ( n ) ( x 0 ) n ! . Example. f ( x ) = 1 3 2 x = 1 1 2( x 1) = n =0 2 n ( x 1) n . R = lim n →∞ 2 n 2 n +1 = 1 2 . 1

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2 Analytic functions Definition 2 A function f ( x ) is analytic at x 0 if f has Taylor series about x 0 which con- verges to f ( x ) in an interval containing x 0 . Example. f ( x ) = 1 1 x is analytic at x = 0; f ( x ) = x is not analytic at 0, since f (0) does not exist; f ( x ) = e x is analytic at any x . Remark: If f and g are analytic at x 0 , then cf , f ± g , fg , f/g (if g ( x 0 ) ̸ = 0) are analytic at x 0 . Remark: If f is analytic at x 0 , then its Taylor series about x 0 is unique. Example: 1 1 x = n =0 x n , | x | < 1 . arctan x = 1 1 + x 2 dx = ( n =0 ( 1) n x 2 n ) dx = x x 3 3 + x 5 5 x 7 7 + ..., 1 < x < 1 . 3 Singular point and ordinary point Consider the equation P ( x ) y ′′ + Q ( x ) y + R ( x ) y = 0 . If we divide two sides by P ( x ), then the equation is changed to y ′′ + p ( x ) y + q ( x ) y = 0 . (1) Definition 3 (Definition 4 in the book) If both p ( x ) and q ( x ) are analytic at a point x 0 , then x 0 is called an ordinary point. Otherwise, it is called a singular point. Example. If p ( x ) and q ( x ) are polynomials, then any point is an ordinary point. Example. The following equation has singular points x = 1 , 2: y ′′ + x + e x x 1 y + x + 1 x 2 y = 0 .
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