Ordinary Differential Equations with Applications to Mechanics

Abstract

PREFACE. INTRODUCTION. Generalities. Ordinary differential equations. Supplementary conditions associated to ODEs. The Cauchy (initial) problem.The two-point problem. 1: LINEAR ODEs OF FIRST AND SECOND ORDER. 1.1 Linear first order ODEs. 1.1.1 Equations of the form . 1.1.2 The linear homogeneous equation. 1.1.3 The general case. 1.1.4 The method of variation of parameters (Lagrange's method). 1.1.5 Differential polynomials. 1.2 Linear second order ODEs. 1.2.1 Homogeneous equations. 1.2.2 Non-homogeneous equations. Lagrange's method. 1.2.3 ODEs with constant coefficients. 1.2.4 Order reduction. 1.2.5 The Cauchy problem. Analytical methods to obtain the solution. 1.2.6 Two-point problems (Picard). 1.2.7 Sturm-Liouville problems. 1.2.8 Linear ODEs of special form. 1.3. Applications 2: LINEAR ODEs OF HIGHER ORDER (n >2). 2.1 The general study of linear ODEs of order . 2.1.1 Generalities. 2.1.2 Linear homogeneous ODEs. 2.1.3 The general solution of the non-homogeneous ODE. 2.1.4 Order reduction. 2.2 Linear ODEs with constant coefficients. 2.2.1 The general solution of the homogeneous equation. 2.2.2 The non-homogeneous ODE. 2.2.3 Euler type ODEs. 2.3 Fundamental solution. Green function. 2.3.1 The fundamental solution. 2.3.2 The Green function. 2.3.3 The non-homogeneous problem. 2.3.4 The homogeneous two-point problem. Eigenvalues. 2.4 Applications. 3: LINEAR ODSs OF FIRST ORDER. 3.1 The general study of linear first order ODSs. 3.1.1 Generalities. 3.1.2 The general solution of the homogeneous ODS. 3.1.3 The general solution of the non-homogeneous ODS. 3.1.4 Order reduction of homogeneous ODSs. 3.1.5 Boundary value problems for ODSs. 3.2 ODSs with constant coefficients. 3.2.1 The general solution of the homogeneous ODS. 3.2.2 Solutions in matrix form for linear ODSs with constant coefficients. 3.3 Applications. 4: NON-LINEAR ODEs OF FIRST AND SECOND ORDER. 4.1 First order non-linear ODEs. 4.1.1 Forms of first order ODEs and oftheir solutions. 4.1.2 Geometric interpretation. The theorem of existence and uniqueness. 4.1.3 Analytic methods for solving first order non-linear ODEs. 4.1.4. First order ODEs integrable by quadratures. 4.2 Non-linear second order ODEs. 4.2.1 Cauchy problems. 4.2.2 Two-point problems. 4.2.3 Order reduction of second order ODEs. 4.2.4 The Bernoulli-Euler equation. 4.2.5 Elliptic integrals. 4.3 Applications. 5: NON-LINEAR ODSs OF FIRST ORDER. 5.1 Generalities. 5.1.1 The general form of a first order ODS. 5.1.2 The existence and uniqueness theorem for the solution of the Cauchy problem. 5.1.3 The particle dynamics. 5.2 First integrals of an ODS. 5.2.1 Generalities. 5.2.2 The theorem of conservation of the kinetic energy. 5.2.3 The symmetric form of an ODS. Integral combinations. 5.2.4 Jacobi's multiplier. The method of the last multiplier. 5.3 Analytical methods of solving the Cauchy problem for non-linear ODSs. 5.3.1 The method of successive approximations (Picard-Lindeloff). 5.3.2 The method of the Taylor series expansion. 5.3.3 The linear equivalence method (LEM). 5.4 Applications. 6: VARIATIONAL CALCULUS. 6.1 Necessary condition of extremum for functionals of integral type. 6.1.1 Generalities. 6.1.2 Functionals of the form..... 6.1.3 Functionals of the form..... 6.1.4 Functionals of integral type, depending on n functions. 6.2 Conditional extrema. 6.2.1 Isoperimetric problems. 6.2.2 Lagrange's problem. 6.3 Applications. 7: STABILITY. 7.1 Lyapunov Stability. 7.1.1 Generalities. 7.1.2 Lyapunov's theorem of stability. 7.2 The stability of the solutions of dynamical systems. 7.2.1 Autonomous dynamical systems. 7.2.2 Long term behaviour of the solutions. 7.3 Applications. INDEX. REFERENCES.