Differential Equations: A Linear Algebra Approach

Cover Page
Title Page
Copyright Page
Dedication
Preface
Table of Contents
1 A Prelude to Differential Equations
    1.1 Introduction
    1.2 Formulation of Differential Equation–Its Significance.
    1.3 Classification of Solutions: General, Particular and Singular Solutions
    1.4 More about Solutions of an ODE
    1.5 Existence-Uniqueness Theorem for Cauchy Problem.
    1.6 Importance of Lipschitz’s condition involved in existence uniqueness theorem in the light of a comparative study between Radiactive decay and Leaky bucket problems:
    1.7 First Order Ode and some of its Qualitative Aspects
2 Equations of First Order and First Degree
    2.1 Introduction
    2.2 Exact Differential Equation
    2.3 Homogeneous Differential Equations
    2.4 Integrating Factor
    2.5 Linear Equations and Bernoulli Equations
    2.6 Integrating Factors Revisited
    2.7 Riccati Equation
    2.8 Application of Differential Equations of First Order
    2.9 Orthogonal Trajectories and Oblique Trajectories
3 A Class of First Order Non-Linear Odes
    3.1 Introduction
    3.2 Non-linear First Order Ode Solvable for p
    3.3 Non-linear Ode Solvable for y
    3.4 Non-linear Ode Solvable for x
    3.5 Existence and Uniqueness Problem
    3.6 Envelopes and Other Loci
    3.7 Clairaut’s Equation and Lagrange’s Equation
4 Linear Algebraic Framework in Differential Equations
    4.1 Introduction
    4.2 Linear Spaces
    4.3 LinearMaps or Transformations
    4.4 Normed Linear Space
    4.5 Bounded Linear Transformation
    4.6 Invertible Operators
5 Differential Equations of Higher Order
    5.1 Introduction
    5.2 Theoretical Aspects
    5.3 Wronskian
    5.4 Working Rules for Homogeneous Linear Ode
    5.5 Few Theoretical Results from Linear Algebra.
    5.6 Symbolic Operator 1/L(D) and Particular Integral
    5.7 Method of Variation of Parameters
    5.8 Special Methods for finding Particular Integrals
    5.9 Method of Undetermined Co-efficients
    5.10 Fourier Series Method for Particular Integrals
6 Second Order Linear Ode: Solution Techniques & Qualitative Analysis
    6.1 Introduction
    6.2 Reduction of Order Method (D’Alembert’s Method)
    6.3 Method of Inspection for finding one Integral
    6.4 Transformation of Second Order Ode by changing the Independent Variable
    6.5 Transformation of a Second order Ode by changing the Dependent Variable:
    6.6 Qualitative Aspects of Second Order Differential Equations
    6.7 Exact Second Order Differential Equations
    6.8 Adjoint Equation and Self-adjoint Odes
    6.9 Sturm-Liouville Problems
    6.10 Green’s Function Approach to IVP
    6.11 Green’s Function Approach to BVP
7 Laplace Transformations in Ordinary Differential Equations
    7.1 Introduction
    7.2 Definition and Anatomy of Laplace Transform
    7.3 Laplace transformation technique of solving Ordinary Differential Equations
8 Series Solutions of Linear Differential Equations
    8.1 Introduction
    8.2 Review of Power Series
    8.3 Solutions about Ordinary Points in the Domain
    8.4 Solution about Regular Singular Points
    8.5 FrobeniusMethod
    8.6 Hypergeometric Equation
    8.7 Irregular singular points
9 Solving Linear Systems by Matrix Methods
    9.1 Introduction
    9.2 Eigenvalue Problems of a square matrix: Diagonalisability
    9.3 Solution of Vector Differential Equation using Eigenvalues of Associated Matrix
    9.4 Operator Norm and Its use in defining Exponentials of Matrices
    9.5 Generalised Eigenvectors: Solution to IVP
    9.6 Jordanization of Matrices and Solution of Ode
    9.7 Fundamental Matrix and Liouville Theorem
    9.8 Alternative Ansatz for Computation of eAx
Appendix
References
Index