7/3/2015 Power Series Method Chapter 5 1. 7/3/2015 Power Series Method Chapter 5 2.

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Presentation transcript:

7/3/2015 Power Series Method Chapter 5 1

7/3/2015 Power Series Method Chapter 5 2

7/3/2015 Power Series Method Chapter 5 3 Note.1. Power series aboutx = 0 is Note.2.Derivatives of Power series

7/3/2015 Power Series Method Chapter 5 4 Note.3.Important Power series (Maclaurin series)

7/3/2015 Power Series Method Chapter Ordinary and Singular points For the linear-second order differential equation Example:Find singular and ordinary points, if any, in the differential equation Note: Solution of the differential equation can be obtained in term of power series about the ordinary point.

7/3/2015 Power Series Method Chapter 5 6 Example:Solve the differential equation. Solution: A series solution of the differential equation about x = 0 as differential equation is analytic about x = 0. by Power series method [1/5]

7/3/2015 Power Series Method Chapter 5 7 Powers of x are same [2/5]

7/3/2015 Power Series Method Chapter 5 8 [3/5]

7/3/2015 Power Series Method Chapter 5 9 [4/5]

7/3/2015 Power Series Method Chapter 5 10 [5/5]

7/3/2015 Power Series Method Chapter 5 11

7/3/2015 Power Series Method Chapter 5 12 Substituting for in the differential equation, we obtained [2/4]

7/3/2015 Power Series Method Chapter 5 13 [3/4]

7/3/2015 Power Series Method Chapter 5 14 where. [4/4]

7/3/2015 Power Series Method Chapter 5 15 Example:3.Solve the differential equation Solution: Differential equation is analytic at x = 0, we can consider a solution in the form of power series. [1/4]

7/3/2015 Power Series Method Chapter 5 16 Substituting in the differential equation, we obtained Letn-2= m n+2 = m m = n+2 m = n-2 Letn-2= m n+2 = m n = m+2 n = m-2 [2/4]

7/3/2015 Power Series Method Chapter 5 17 [3/4]

7/3/2015 Power Series Method Chapter 5 18 substituting values of C’s in series [4/4]

7/3/2015 Power Series Method Chapter 5 19

7/3/2015 Power Series Method Chapter 5 20

7/3/2015 Power Series Method Chapter 5 21

7/3/2015 Power Series Method Chapter 5 22

7/3/2015 Power Series Method Chapter 5 23

7/3/2015 Power Series Method Chapter 5 24

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7/3/2015 Power Series Method Chapter 5 30