Book contents
- Frontmatter
- Contents
- Preface
- I MATRIX THEORY
- III VECTORS AND TENSORS
- III ORDINARY DIFFERENTIAL EQUATIONS
- 6 Analytical Solutions of Ordinary Differential Equations
- 7 Numerical Solution of Initial and Boundary Value Problems
- 8 Qualitative Analysis of Ordinary Differential Equations
- 9 Series Solutions of Linear Ordinary Differential Equations
- IV PARTIAL DIFFERENTIAL EQUATIONS
- A Additional Details and Fortification for Chapter 1
- B Additional Details and Fortification for Chapter 2
- C Additional Details and Fortification for Chapter 3
- D Additional Details and Fortification for Chapter 4
- E Additional Details and Fortification for Chapter 5
- F Additional Details and Fortification for Chapter 6
- G Additional Details and Fortification for Chapter 7
- H Additional Details and Fortification for Chapter 8
- I Additional Details and Fortification for Chapter 9
- J Additional Details and Fortification for Chapter 10
- K Additional Details and Fortification for Chapter 11
- L Additional Details and Fortification for Chapter 12
- M Additional Details and Fortification for Chapter 13
- N Additional Details and Fortification for Chapter 14
- Bibliography
- Index
9 - Series Solutions of Linear Ordinary Differential Equations
from III - ORDINARY DIFFERENTIAL EQUATIONS
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Preface
- I MATRIX THEORY
- III VECTORS AND TENSORS
- III ORDINARY DIFFERENTIAL EQUATIONS
- 6 Analytical Solutions of Ordinary Differential Equations
- 7 Numerical Solution of Initial and Boundary Value Problems
- 8 Qualitative Analysis of Ordinary Differential Equations
- 9 Series Solutions of Linear Ordinary Differential Equations
- IV PARTIAL DIFFERENTIAL EQUATIONS
- A Additional Details and Fortification for Chapter 1
- B Additional Details and Fortification for Chapter 2
- C Additional Details and Fortification for Chapter 3
- D Additional Details and Fortification for Chapter 4
- E Additional Details and Fortification for Chapter 5
- F Additional Details and Fortification for Chapter 6
- G Additional Details and Fortification for Chapter 7
- H Additional Details and Fortification for Chapter 8
- I Additional Details and Fortification for Chapter 9
- J Additional Details and Fortification for Chapter 10
- K Additional Details and Fortification for Chapter 11
- L Additional Details and Fortification for Chapter 12
- M Additional Details and Fortification for Chapter 13
- N Additional Details and Fortification for Chapter 14
- Bibliography
- Index
Summary
In this chapter, we focus our attention on obtaining analytical solutions of linear differential equations with coefficients that are not constant. These solutions are not as simple as those for which the coefficients were constant. One general approach is to use a power series formulation.
In Section 9.1, we describe the main approaches of power series solution. Depending on the equation, one can choose to expand the solution around an ordinary point or a singular point. Each of these choices will determine the structure of the series. For an ordinary point, the expansion is simply a Taylor series, whereas for a singular point, we need a series known as a Frobenius series.
Although the power series method is straightforward, power series solutions can be quite lengthy and complicated. Nonetheless, for certain equations, solutions can be found based on the parameters of the equations, thus yielding direct solutions. This is the case for two important classes of second-order equations that have several applications. These are the Legendre equations and Bessel equations, which we describe in Sections 9.2 and 9.3, respectively.
We have also included other important equations in the exercises, such as hyper-geometric equations, Jacobi equations, Laguerre equations, Hermite equations, and so forth, where the same techniques given in this chapter can be used to generate the useful functions and polynomials. Fortunately, the special functions and polynomials that solve these equations, including Legendre polynomials, Legendre functions and Bessel functions, are included in several computer software programs such as MATLAB.
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- Information
- Methods of Applied Mathematics for Engineers and Scientists , pp. 347 - 376Publisher: Cambridge University PressPrint publication year: 2013