Ordinary Differential Equations and Special Functions

Chapter 1: Preliminaries I

• • 1.1 Introduction
• • 1.2 Formation of a Differential Equation
• • 1.3 Family of Curves Represented by Ordinary Differential Equations
• • 1.4 Equation of the First Order and First Degree
• • 1.5 Equations of the First Order and Higher Degree
• • 1.6 Linear Differential Equation
• • 1.7 Other Methods of Finding P.I.
• • 1.8 Differential Equation of Other Types
• • 1.9 Orthogonal Trajectories
• • 1.10 Examples
• • 1.11 Exercise

Chapter 2: Existence Theorems

• • 2.1 Introduction
• • 2.2 Initial Value Problems and Boundary Value Problems
• • 2.3 Picard's Method of Successive Approximation
• • 2.4 Lipschitz Condition
• • 2.5 Picard's Theorem: Existence and Uniqueness Theorem
• • 2.6 Singular Solutions
• • 2.7 Clairaut Equation
• • 2.8 Examples
• • 2.9 Exercise

Chapter 3: System of Linear Differential Equations-I

• • 3.1 Introduction
• • 3.2 Matrix Form of a Linear System
• • 3.3 Reduction of an nth-Order Equation
• • 3.4 Matrix Preliminaries
• • 3.5 Fundamental Set of Solutions
• • 3.6 Solution of Non-Homogeneous Linear Systems
• • 3.7 Linear System with Constant Coefficients
• • 3.8 Exercise

Chapter 4: Systems of Linear Differential Equations-II

• • 4.1 Introduction
• • 4.2 Linearly Dependent and Independent Functions
• • 4.3 The Second-Order Homogeneous Equation
• • 4.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters
• • 4.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients
• • 4.6 Examples
• • 4.7 Exercise

Chapter 5: Adjoint Equation

• • 5.1 Introduction
• • 5.2 Adjoint Equation
• • 5.3 Green's Formula
• • 5.4 Examples
• • 5.5 Exercise

Chapter 6: Boundary Value Problem

• • 6.1 Introduction
• • 6.2 Green's Function
• • 6.3 Examples
• • 6.4 Exercise

Chapter 7: Strum Liouville Problem

• • 7.1 Introduction
• • 7.2 Strum–Liouville Equation
• • 7.3 Orthogonality of Eigen Functions
• • 7.4 Orthonormal Set of Functions
• • 7.5 Gram–Schmidt Process of Orthonormalization
• • 7.6 Reality of Eigenvalues
• • 7.7 Examples
• • 7.8 Exercise

Chapter 8: Preliminaries II

• • 8.1 Introduction
• • 8.2 Infinite Series
• • 8.3 Infinite Integrals
• • 8.4 Infinite Products
• • 8.5 Some Theorems on Functions of Complex Variables
• • 8.6 Exercise

Chapter 9: Series Solution of Differential Equations

• • 9.1 Introduction
• • 9.2 Power Series
• • 9.3 Power Series Solution Near the Ordinary Point x = x0
• • 9.4 Series Solution About Regular Singular Point x = 0: Frobenius Method
• • 9.5 Examples
• • 9.6 Exercise

Chapter 10: Hypergeometric Functions

• • 10.1 Introduction
• • 10.2 Differentiation of Hypergeometric Functions
• • 10.3 An Integral Formula for a Hypergeometric Function
• • 10.4 Transformation of F (α,β ,γ ;x)
• • 10.5 Hypergeometric Equation
• • 10.6 Confluent Hypergeometric Series
• • 10.7 Contiguous Hypergeometric Functions
• • 10.8 Generalized Hypergeometric Series
• • 10.9 Integrals Involving Generalized Hypergeometric Functions
• • 10.10 Some Special Generalized Hypergeometric Functions
• • 10.11 Barnes Type Contour Integrals
• • 10.12 Example
• • 10.13 Exercise

Chapter 11: Bessel Functions

• • 11.1 Introduction
• • 11.2 Bessel's Equation
• • 11.3 Recurrence Formulae for Jn(x)
• • 11.4 Expansion of J0 , J1 ,and J1/2
• • 11.5 Generating Function for Jn(x)
• • 11.6 Modified Bessel Functions
• • 11.7 Equations Reducible to Bessel Equation
• • 11.8 Orthogonality of Bessel Functions
• • 11.9 Zeros of Bessel Functions
• • 11.10 Ber and Bei Functions
• • 11.11 Exercise

Chapter 12: Legendre Polynomials

• • 12.1 Introduction
• • 12.2 Legendre's Equation
• • 12.3 Another Form of Legendre's Polynomial Pn(x)
• • 12.4 Generating Function for Legendre's Polynomials
• • 12.5 Various Forms of Pn(x)
• • 12.6 Recurrence Formulae for Pn(x)
• • 12.7 Christoffel's Summation Formula
• • 12.8 Orthogonality of Legendre Polynomials
• • 12.9 Fourier–Legendre's Expansion of f (x)
• • 12.10 Associated Legendre's Functions
• • 12.11 Legendre's Functions of the Second Kind—Qn(x)
• • 12.12 Examples
• • 12.13 Exercise

Chapter 13: Hermite Polynomials

• • 13.1 Introduction
• • 13.2 Hermite Equation and Its Solution
• • 13.3 Generating Function for Hermite Polynomials
• • 13.4 Recurrence Relations
• • 13.5 Orthogonal Property
• • 13.6 Expansion of Polynomials
• • 13.7 More Generating Functions
• • 13.8 Examples
• • 13.9 Exercise

Chapter 14: Laguerre Polynomials

• • 14.1 Introduction
• • 14.2 Laguerre's Equation and Its Solution
• • 14.3 Generating Function of Laguerre Polynomials
• • 14.4 Orthogonality Properties of Laguerre Polynomials
• • 14.5 Recurrence Relations
• • 14.6 Expansion of Laguerre Polynomials
• • 14.7 Properties of Laguerre Polynomials
• • 14.8 Generalized Laguerre Polynomial
• • 14.9 Examples
• • 14.10 Exercise

Chapter 15: Jacobi Polynomials

• • 15.1 Introduction
• • 15.2 Jacobi Polynomial
• • 15.3 Generating Functions
• • 15.4 Rodrigues' Formula
• • 15.5 Orthogonality of Jacobi Polynomial
• • 15.6 Recurrence Relations
• • 15.7 Expansions
• • 15.8 Examples
• • 15.9 Exercise

Chapter 16: Chebyshev Polynomials

• • 16.1 Introduction
• • 16.2 Chebyshev Polynomials
• • 16.3 Orthogonality Property
• • 16.4 Recurrence Relations
• • 16.5 Identities of Chebyshev Polynomials
• • 16.6 Expansions
• • 16.7 Generating Function
• • 16.8 Rodrigues Formula of Chebyshev Polynomials
• • 16.9 Exercise

Chapter Appendix A: Answer to Even-Numbered Exercises