THEORY OF ORDINARY DIFFERENTIAL EQUATIONS

Transcription

1 Introduction to THEORY OF ORDINARY DIFFERENTIAL EQUATIONS V. Dharmaiah

2 Contents i Introduction to Theory of Ordinary Differential Equations

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4 Introduction to Theory of Ordinary Differential Equations V. Dharmaiah Department of Mathematics Osmania University Hyderabad Delhi

5 Introduction to Theory of Ordinary Differential Equations V. Dharmaiah 2013 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi and Printed by Mohan Makhijani at Rekha Printers Private Limited, New Delhi

6 To My elder brother Endavetla Devaiah

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8 Contents Preface xi 1. Preliminaries Introduction Basic Concepts Formation of Differential Equations by Elimination of Arbitrary Constants Classification of Solutions Geometrical Interpretation of a Differential Equation of First Order and First Degree Geometrical Interpretation of a Differential Equation of First Order and Higher Degree Geometrical Interpretation of a Differential Equation of Second and Higher Order 7 2. Existence Theorems Introduction Initial Value Problems and Boundary Value Problems Lipschitz Condition Gronwall s Integral Inequality Integral Equations Equivalent to IVPs Successive Approximations Existence Theorem Nonlocal Existence Theorem Approximations to, and Uniqueness of, Solutions Fixed Point Method Continuation of Solutions Dependence of Solutions on Initial Conditions Equations with Complex-Valued Functions Analytical Methods Introduction Existence Theorem Extremal Solutions Lower and Upper Solutions 55 vii

9 viii Contents 3.5 Monotone Iterative Method and Method of Quasilinearization Bihari s Integral Inequality First Order Equations Introduction Exact Equations Equations Reducible to Exact Form Integrating Factors Found by Inspection Variable Separable Equations Equations Reducible to Variable Separable Form Linear Equations of First Order Equations Reducible to Linear Form Equations of First Order and Higher Degree Systems of Equations Introduction Complex n-dimensional Space Lipschitz Condition Systems as Vector Equations Existence and Uniqueness Theorems for Systems Linear Systems in Vector Variables Existence Theorems for Equations of Order n Linear Systems Introduction Matrix Operations Linear Systems in Vector-Matrix Form Modelling Problems Homogeneous Linear Systems Fundamental Matrix Solutions of Nonhomogeneous Linear Systems Homogeneous Linear Systems with Constant Coefficients Periodic Solutions of Linear Systems Linear Equations with Variable Coefficients Introduction Initial Value Problems (IVPs) for Linear Equations of Order n Solutions of Homogeneous Equations Wronskian and Linear Independence Reduction of Order Nonhomogeneous Equations Homogeneous Equations with Analytic Coefficients 213

10 Contents ix 8. Linear Equations with Constant Coefficients Introduction Homogeneous Equations of Order n Linear Equations with Real Coefficients Algebra of Constant Coefficient Operators Annihilator Method or Method of Undetermined Coefficients Linear Equations with Regular Singular Points Introduction Euler Equation Cauchy Equation Linear Equations of Order Two with Variable Coefficients and with Regular Singular Point at x 0 = Regular Singular Points at Infinity Generalization of variation of Parameters Formula Generalization of Variation of Parameters Formula Adjoint Equations Introduction Adjoint Equations for Linear Equations of Order One Adjoint Equations for Linear Equations of Order Two Riccati Equation Riccati Equation Transforming Riccati Equation Solutions of Riccati Equation Boundary Value Problems Introduction Sturm Liouville Problems Green s Function Non-existence of Solutions Existence Theorem for Nonlinear Differential Equations Oscillation Theory (For Linear Differential Equations of Order Two) Introduction Number of Zeros in a Finite Interval Sturm Separation Theorem Sturm Comparison Theorem Sturm Picone Theorem 325

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