{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}