AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS

By N. LITIA

CONTENTS INTRODUCTION PART I FIRST ORDER DIFFERENTIAL EQUATIONS RADIOACTIVE DECAY AND CARBON DATING RADIOACTIVE DECAY RADIOCARBON DATING EXERCISES INTEGRATION VARIABLES CLASSI?CATION OF DIFFERENTIAL EQUATIONS ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS THE ORDER OF A DIFFERENTIAL EQUATION LINEAR AND NONLINEAR DIFFERENT TYPES OF SOLUTION *GRAPHICAL REPRESENTATION OF SOLUTIONS USING MATLAB ‘TRIVIAL’ DIFFERENTIAL EQUATIONS THE FUNDAMENTAL THEOREM OF CALCULUS GENERAL SOLUTIONS AND INITIAL CONDITIONS VELOCITY, ACCELERATION AND NEWTON’S SECOND LAW OF MOTION AN EQUATION THAT WE CANNOT SOLVE EXPLICITLY EXISTENCE AND UNIQUENESS OF SOLUTIONS THE CASE FOR AN ABSTRACT RESULT THE EXISTENCE AND UNIQUENESS THEOREM MAXIMAL INTERVAL OF EXISTENCE THE CLAY MATHEMATICS INSTITUTE’S $1 000 QUESTION SCALAR AUTONOMOUS ODES THE QUALITATIVE APPROACH STABILITY, INSTABILITY AND BIFURCATION ANALYTIC CONDITIONS FOR STABILITY AND INSTABILITY STRUCTURAL STABILITY AND BIFURCATIONS SOME EXAMPLES THE PITCHFORK BIFURCATION DYNAMICAL SYSTEMS SEPARABLE EQUATIONS THE SOLUTION ‘RECIPE’ THE LINEAR EQUATION ?X = ?X MALTHUS’ POPULATION MODEL JUSTIFYING THE METHOD A MORE REALISTIC POPULATION MODEL FURTHER EXAMPLES FIRST ORDER LINEAR EQUATIONS AND THE INTEGRATING FACTOR CONSTANT COEF?CIENTS INTEGRATING FACTORS EXAMPLES NEWTON’S LAW OF COOLING TWO ‘TRICKS’ FOR NONLINEAR EQUATIONS EXACT EQUATIONS SUBSTITUTION METHODS PART II SECOND ORDER LINEAR EQUATIONS WITH CONSTANT COEF?CIENTS SECOND ORDER LINEAR EQUATIONS: GENERAL THEORY EXISTENCE AND UNIQUENESS LINEARITY LINEARLY INDEPENDENT SOLUTIONS *THE WRONSKIAN *LINEAR ALGEBRA HOMOGENEOUS SECOND ORDER LINEAR EQUATIONS TWO DISTINCT REAL ROOTS A REPEATED REAL ROOT NO REAL ROOTS OSCILLATIONS THE SPRING THE SIMPLE PENDULUM DAMPED OSCILLATIONS INHOMOGENEOUS SECOND ORDER LINEAR EQUATIONS COMPLEMENTARY FUNCTION AND PARTICULAR INTEGRAL WHEN F (T) IS A POLYNOMIAL WHEN F (T) IS AN EXPONENTIAL WHEN F (T) IS A SINE OR COSINE RULE OF THUMB MORE COMPLICATED FUNCTIONS F (T) RESONANCE PERIODIC FORCING PSEUDO RESONANCE IN PHYSICAL SYSTEMS HIGHER ORDER LINEAR EQUATIONS *THE GENERAL THEORY FOR NTH ORDER EQUATIONS PART III LINEAR SECOND ORDER EQUATIONS WITH VARIABLE COEF?CIENTS REDUCTION OF ORDER *THE VARIATION OF CONSTANTS FORMULA *CAUCHY–EULER EQUATIONS TWO REAL ROOTS A REPEATED ROOT COMPLEX ROOTS *SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS POWER SERIES ORDINARY POINTS REGULAR SINGULAR POINTS BESSEL’S EQUATION PART IV NUMERICAL METHODS AND DIFFERENCE EQUATIONS EULER’S METHOD AN EXAMPLE *MATLAB IMPLEMENTATION OF EULER’S METHOD CONVERGENCE OF EULER’S METHOD DIFFERENCE EQUATIONS FIRST ORDER DIFFERENCE EQUATIONS SECOND ORDER DIFFERENCE EQUATIONS THE HOMOGENEOUS EQUATION PARTICULAR SOLUTIONS NONLINEAR ?RST ORDER DIFFERENCE EQUATIONS FIXED POINTS AND STABILITY COBWEB DIAGRAMS PERIODIC ORBITS EULER’S METHOD FOR AUTONOMOUS EQUATIONS THE LOGISTIC MAP FIXED POINTS AND THEIR STABILITY THE PERIOD-DOUBLING CASCADE THE BIFURCATION DIAGRAM AND MORE PERIODIC ORBITS CHAOS *ANALYSIS OF XN+1 = 4XN(1 ?XN) PART V COUPLED LINEAR EQUATIONS *VECTOR ?RST ORDER EQUATIONS AND HIGHER ORDER EQUATIONS EXISTENCE AND UNIQUENESS FOR SECOND ORDER EQUATIONS EXPLICIT SOLUTIONS OF COUPLED LINEAR SYSTEMS EIGENVALUES AND EIGENVECTORS REWRITING THE EQUATION IN MATRIX FORM *EIGENVALUES AND EIGENVECTORS WITH MATLAB DISTINCT REAL EIGENVALUES THE EXPLICIT SOLUTION CHANGING COORDINATES PHASE DIAGRAMS FOR UNCOUPLED EQUATIONS PHASE DIAGRAMS FOR COUPLED EQUATIONS STABLE AND UNSTABLE MANIFOLDS COMPLEX EIGENVALUES CHANGING COORDINATES AND THE PHASE PORTRAIT THE PHASE PORTRAIT FOR THE ORIGINAL EQUATION A REPEATED REAL EIGENVALUE A IS A MULTIPLE OF THE IDENTITY: STARS A IS NOT A MULTIPLE OF THE IDENTITY: IMPROPER NODES SUMMARY OF PHASE PORTRAITS FOR LINEAR EQUATIONS *JORDAN CANONICAL FORM PART VI COUPLED NONLINEAR EQUATIONS COUPLED NONLINEAR EQUATIONS SOME COMMENTS ON PHASE PORTRAITS COMPETITION OF SPECIES DIRECTION ?ELDS ANALYTICAL METHOD FOR PHASE PORTRAITS ECOLOGICAL MODELS COMPETING SPECIES PREDATOR-PREY MODELS I PREDATOR-PREY MODELS II NEWTONIAN DYNAMICS ONE-DIMENSIONAL CONSERVATIVE SYSTEMS *A BEAD ON A WIRE DISSIPATIVE SYSTEMS THE ‘REAL’ PENDULUM THE UNDAMPED PENDULUM THE DAMPED PENDULUM ALTERNATIVE PHASE SPACE *PERIODIC ORBITS DULAC’S CRITERION THE POINACR´E–BENDIXSON THEOREM *THE LORENZ EQUATIONS WHAT NEXT? PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS DYNAMICAL SYSTEMS AND CHAOS