MATHEMATICS: AN INTRODUCTION TO DIFFERENTIAL EQUATIONS

By MICHAEL LEE

CONTENTS 1 DI?ERENTIAL EQUATIONS AND MODELS DI?ERENTIAL EQUATIONS EQUATIONS AND SOLUTIONS GEOMETRICAL INTERPRETATION PURE TIME EQUATIONS MATHEMATICAL MODELS PARTICLE DYNAMICS AUTONOMOUS DI?ERENTIAL EQUATIONS STABILITY AND BIFURCATION HEAT TRANSFER CHEMICAL REACTORS ELECTRIC CIRCUITS 2 ANALYTIC SOLUTIONS AND APPROXIMATIONS SEPARATION OF VARIABLES FIRST-ORDER LINEAR EQUATIONS APPROXIMATION PICARD ITERATION NUMERICAL METHODS ERROR ANALYSIS CONTENTS 3 SECOND-ORDER DI?ERENTIAL EQUATIONS PARTICLE MECHANICS LINEAR EQUATIONS WITH CONSTANT COE?CIENTS THE NONHOMOGENEOUS EQUATION UNDETERMINED COE?CIENTS RESONANCE VARIABLE COE?CIENTS CAUCHY–EULER EQUATION POWER SERIES SOLUTIONS REDUCTION OF ORDER VARIATION OF PARAMETERS BOUNDARY VALUE PROBLEMS AND HEAT FLOW HIGHER-ORDER EQUATIONS SUMMARY AND REVIEW 4 LAPLACE TRANSFORMS DE?NITION AND BASIC PROPERTIES INITIAL VALUE PROBLEMS THE CONVOLUTION PROPERTY DISCONTINUOUS SOURCES POINT SOURCES TABLE OF LAPLACE TRANSFORMS 5 LINEAR SYSTEMS INTRODUCTION MATRICES TWO-DIMENSIONAL SYSTEMS SOLUTIONS AND LINEAR ORBITS THE EIGENVALUE PROBLEM REAL UNEQUAL EIGENVALUES COMPLEX EIGENVALUES REAL, REPEATED EIGENVALUES STABILITY NONHOMOGENEOUS SYSTEMS THREE-DIMENSIONAL SYSTEMS 6 NONLINEAR SYSTEMS NONLINEAR MODELS PHASE PLANE PHENOMENA THE LOTKA–VOLTERRA MODEL HOLLING FUNCTIONAL RESPONSES AN EPIDEMIC MODEL LINEARIZATION AND STABILITY PERIODIC SOLUTIONS THE POINCAR´E–BENDIXSON THEOREM