ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATIONS TO MECHANICS

By PATRICIA MARTINEZ

CONTENTS INTRODUCTION 1 GENERALITIES 2 ORDINARY DIFFERENTIAL EQUATIONS 3 SUPPLEMENTARY CONDITIONS 3 1 THE CAUCHY (INITIAL) PROBLEM 3 2 THE TWO-POINT PROBLEM 1 LINEAR ODES OF FIRST AND SECOND ORDER 1 LINEAR FIRST ORDER ODES 1 1 EQUATIONS OF THE FORM ( ) X F Y = ? 1 2 THE LINEAR HOMOGENEOUS EQUATION 1 3 THE GENERAL CASE 1 4 THE METHOD OF VARIATION OF PARAMETERS (LAGRANGE’S METHOD) 1 5 DIFFERENTIAL POLYNOMIALS 2 LINEAR SECOND ORDER ODES 2 1 HOMOGENEOUS EQUATIONS 2 2 NON-HOMOGENEOUS EQUATIONS LAGRANGE’S METHOD 2 3 ODES WITH CONSTANT COEFFICIENTS 2 4 ORDER REDUCTION 2 5 THE CAUCHY PROBLEM ANALYTICAL METHODS TO OBTAIN THE SOLUTION 2 6 TWO-POINT PROBLEMS (PICARD) 2 7 STURM-LIOUVILLE PROBLEMS 2 8 LINEAR ODES OF SPECIAL FORM 3 APPLICATIONS 2 LINEAR ODES OF HIGHER ORDER ( > N ) 1 THE GENERAL STUDY OF LINEAR ODES OF ORDER > N 1 1 GENERALITIES 1 2 LINEAR HOMOGENEOUS ODES 1 3 THE GENERAL SOLUTION OF THE NON-HOMOGENEOUS ODE 1 4 ORDER REDUCTION 2 LINEAR ODES WITH CONSTANT COEFFICIENTS 2 1 THE GENERAL SOLUTION OF THE HOMOGENEOUS EQUATION 2 2 THE NON-HOMOGENEOUS ODE 2 3 EULER TYPE ODES 3 FUNDAMENTAL SOLUTION GREEN FUNCTION 3 1 THE FUNDAMENTAL SOLUTION 3 2 THE GREEN FUNCTION ASSOCIATED TO ODES 3 LINEAR ODSS OF FIRST ORDER 1 THE GENERAL STUDY OF LINEAR FIRST ORDER ODSS 1 2 THE GENERAL SOLUTION OF THE HOMOGENEOUS ODS 1 3 THE GENERAL SOLUTION OF THE NON-HOMOGENEOUS ODS 1 4 ORDER REDUCTION OF HOMOGENEOUS ODSS 1 5 BOUNDARY VALUE PROBLEMS FOR ODSS 2 ODSS WITH CONSTANT COEFFICIENTS 2 1 THE GENERAL SOLUTION OF THE HOMOGENEOUS ODS 2 2 SOLUTIONS IN MATRIX FORM FOR LINEAR ODSS WITH CONSTANT COEFFICIENTS 4 NON-LINEAR ODES OF FIRST AND SECOND ORDER 1 FIRST ORDER NON-LINEAR ODES 1 1 FORMS OF FIRST ORDER ODES AND OF THEIR SOLUTIONS 1 1 1 FORMS OF ODES 1 1 2 FORMS OF THE SOLUTIONS 1 2 GEOMETRIC INTERPRETATION THE THEOREM OF EXISTENCE AND UNIQUENESS 1 3 ANALYTIC METHODS FOR SOLVING FIRST ORDER NON-LINEAR ODES 1 4 FIRST ORDER ODES INTEGRABLE BY QUADRATURES 1 4 1 ODES WITH SEPARATE VARIABLES 1 4 2 ODES WITH SEPARABLE VARIABLES 1 4 3 HOMOGENEOUS FIRST ORDER ODES 1 4 4 ODES OF THE FORM 1 4 5 TOTAL DIFFERENTIAL ODES 1 4 6 INTEGRANT FACTOR 1 4 7 CLAIRAUT’S EQUATION 1 4 8 LAGRANGE’S EQUATION 1 4 9 BERNOULLI’S EQUATION 1 4 10 RICCATI’S EQUATION 2 NON-LINEAR SECOND ORDER ODES 2 1 CAUCHY PROBLEMS 2 2 TWO-POINT PROBLEMS 2 3 ORDER REDUCTION OF SECOND ORDER ODES 2 4 THE BERNOULLI-EULER EQUATION 2 5 ELLIPTIC INTEGRALS 3 3 THE NON-HOMOGENEOUS PROBLEM 3 4 THE HOMOGENEOUS TWO-POINT PROBLEM EIGENVALUES 4 APPLICATIONS 2 FIRST INTEGRALS OF AN ODS 2 1 GENERALITIES 2 2 THE THEOREM OF CONSERVATION OF THE KINETIC ENERGY 2 3 THE SYMMETRIC FORM OF AN ODS INTEGRAL COMBINATIONS 2 4 JACOBI’S MULTIPLIER THE METHOD OF THE LAST MULTIPLIER 3 ANALYTICAL METHODS OF SOLVING THE CAUCHY PROBLEM FOR NON-LINEAR ODSS 3 1 THE METHOD OF SUCCESSIVE APPROXIMATIONS (PICARD-LINDELÕFF) 3 2 THE METHOD OF THE TAYLOR SERIES EXPANSION 3 3 THE LINEAR EQUIVALENCE METHOD (LEM) 3 3 1 SOLUTIONS OF NON-LINEAR ODSS BY LEM 3 3 2 NEW LEM REPRESENTATIONS IN THE CASE OF POLYNOMIAL COEFFICIENTS 4 APPLICATIONS 6 VARIATIONAL CALCULUS 1 NECESSARY CONDITION OF EXTREMUM FOR FUNCTIONALS OF INTEGRAL TYPE 1 2 FUNCTIONALS OF THE FORM [ ] ( ) ( ) ( ) X X Y X Y X F Y X X D , , I 1 3 FUNCTIONALS OF THE FORM [ ] ( ) ( ) X Y Y Y Y X F Y N X X D ,, , , , I 1 4 FUNCTIONALS OF INTEGRAL TYPE, DEPENDING ON N FUNCTIONS 2 CONDITIONAL EXTREMA 2 1 ISOPERIMETRIC PROBLEMS 2 2 LAGRANGE’S PROBLEM 7 STABILITY 1 LYAPUNOV STABILITY 1 2 LYAPUNOV’S THEOREM OF STABILITY 5 NON-LINEAR ODSS OF FIRST ORDER 1 1 THE GENERAL FORM OF A FIRST ORDER ODS 1 2 THE EXISTENCE AND UNIQUENESS THEOREM FOR THE SOLUTION OF THE CAUCHY PROBLEM 1 3 THE PARTICLE DYNAMICS 2 THE STABILITY OF THE SOLUTIONS OF DYNAMICAL SYSTEMS 2 1 AUTONOMOUS DYNAMICAL SYSTEMS 2 2 LONG TERM BEHAVIOUR OF THE SOLUTIONS REFERENCES