DIFFERENTIAL EQUATIONS

By Q. MIARO

CONTENTS PART ONE: LINEAR EQUATIONS VARIABLE COE?CIENT, SECOND ORDER, LINEAR, ORDINARY DI?ERENTIAL EQUATIONS THE METHOD OF REDUCTION OF ORDER THE METHOD OF VARIATION OF PARAMETERS SOLUTION BY POWER SERIES: THE METHOD OF FROBENIUS LEGENDRE FUNCTIONS DE?NITION OF THE LEGENDRE POLYNOMIALS, PN(X) THE GENERATING FUNCTION FOR PN(X) DI?ERENTIAL AND RECURRENCE RELATIONS BETWEEN LEGENDRE POLYNOMIALS RODRIGUES’ FORMULA ORTHOGONALITY OF THE LEGENDRE POLYNOMIALS PHYSICAL APPLICATIONS OF THE LEGENDRE POLYNOMIALS THE ASSOCIATED LEGENDRE EQUATION BESSEL FUNCTIONS THE GAMMA FUNCTION AND THE POCKHAMMER SYMBOL SERIES SOLUTIONS OF BESSEL’S EQUATION THE GENERATING FUNCTION FOR JN(X), N AN INTEGER DI?ERENTIAL AND RECURRENCE RELATIONS BETWEEN BESSEL FUNCTIONS MODI?ED BESSEL FUNCTIONS ORTHOGONALITY OF THE BESSEL FUNCTIONS INHOMOGENEOUS TERMS IN BESSEL’S EQUATION SOLUTIONS EXPRESSIBLE AS BESSEL FUNCTIONS PHYSICAL APPLICATIONS OF THE BESSEL FUNCTIONS BOUNDARY VALUE PROBLEMS, GREEN’S FUNCTIONS AND STURM–LIOUVILLE THEORY INHOMOGENEOUS LINEAR BOUNDARY VALUE PROBLEMS THE SOLUTION OF BOUNDARY VALUE PROBLEMS BY EIGENFUNCTION EXPANSIONS STURM–LIOUVILLE SYSTEMS FOURIER SERIES AND THE FOURIER TRANSFORM GENERAL FOURIER SERIES THE FOURIER TRANSFORM GREEN’S FUNCTIONS REVISITED SOLUTION OF LAPLACE’S EQUATION USING FOURIER TRANSFORMS GENERALIZATION TO HIGHER DIMENSIONS LAPLACE TRANSFORMS DE?NITION AND EXAMPLES PROPERTIES OF THE LAPLACE TRANSFORM THE SOLUTION OF ORDINARY DI?ERENTIAL EQUATIONS USING LAPLACE TRANSFORMS THE INVERSION FORMULA FOR LAPLACE TRANSFORMS CLASSI?CATION, PROPERTIES AND COMPLEX VARIABLE METHODS FOR SECOND ORDER PARTIAL DI?ERENTIAL EQUATIONS CLASSI?CATION AND PROPERTIES OF LINEAR, SECOND ORDER PARTIAL DI?ERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES COMPLEX VARIABLE METHODS FOR SOLVING LAPLACE’S EQUATION PART TWO: NONLINEAR EQUATIONS AND ADVANCED TECHNIQUES EXISTENCE, UNIQUENESS, CONTINUITY AND COMPARISON OF SOLUTIONS OF ORDINARY DI?ERENTIAL EQUATIONS LOCAL EXISTENCE OF SOLUTIONS UNIQUENESS OF SOLUTIONS DEPENDENCE OF THE SOLUTION ON THE INITIAL CONDITIONS COMPARISON THEOREMS NONLINEAR ORDINARY DI?ERENTIAL EQUATIONS: PHASE PLANE METHODS INTRODUCTION: THE SIMPLE PENDULUM FIRST ORDER AUTONOMOUS NONLINEAR ORDINARY DI?ERENTIAL EQUATIONS SECOND ORDER AUTONOMOUS NONLINEAR ORDINARY DI?ERENTIAL EQUATIONS THIRD ORDER AUTONOMOUS NONLINEAR ORDINARY DI?ERENTIAL EQUATIONS GROUP THEORETICAL METHODS LIE GROUPS INVARIANTS UNDER GROUP ACTION THE EXTENDED GROUP INTEGRATION OF A FIRST ORDER EQUATION WITH A KNOWN GROUP INVARIANT TOWARDS THE SYSTEMATIC DETERMINATION OF GROUPS UNDER WHICH A FIRST ORDER EQUATION IS INVARIANT INVARIANTS FOR SECOND ORDER DI?ERENTIAL EQUATIONS PARTIAL DI?ERENTIAL EQUATIONS ASYMPTOTIC METHODS: BASIC IDEAS ASYMPTOTIC EXPANSIONS THE ASYMPTOTIC EVALUATION OF INTEGRALS ASYMPTOTIC METHODS: DI?ERENTIAL EQUATIONS AN INSTRUCTIVE ANALOGY: ALGEBRAIC EQUATIONS ORDINARY DI?ERENTIAL EQUATIONS STABILITY, INSTABILITY AND BIFURCATIONS ZERO EIGENVALUES AND THE CENTRE MANIFOLD THEOREM LYAPUNOV’S THEOREMS BIFURCATION THEORY TIME-OPTIMAL CONTROL IN THE PHASE PLANE DE?NITIONS FIRST ORDER EQUATIONS SECOND ORDER EQUATIONS EXAMPLES OF SECOND ORDER CONTROL PROBLEMS PROPERTIES OF THE CONTROLLABLE SET THE CONTROLLABILITY MATRIX THE TIME-OPTIMAL MAXIMUM PRINCIPLE (TOMP) AN INTRODUCTION TO CHAOTIC SYSTEMS THREE SIMPLE CHAOTIC SYSTEMS MAPPINGS THE POINCAR´E RETURN MAP HOMOCLINIC TANGLES QUANTIFYING CHAOS: LYAPUNOV EXPONENTS AND THE LYAPUNOV SPECTRUM