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9781402054396

Ordinary Differential Equations With Applications to Mechanics

by ; ;
  • ISBN13:

    9781402054396

  • ISBN10:

    1402054394

  • Format: Hardcover
  • Copyright: 2007-01-03
  • Publisher: Springer Verlag
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Summary

The present book has its source in the authors' wish to create a bridge between the mathematical and the technical disciplines, which need a good knowledge of a strong mathematical tool. The necessity of such an interdisciplinary work drove the authors to publish a first book to this aim with Editura Tehnica, Bucharest, Romania.The present book is a new, English edition of the volume published in 1999. It contains many improvements concerning the theoretical (mathematical) information, as well as new topics, using enlarged and updated references. Only ordinary differential equations and their solutions in an analytical frame were considered, leaving aside their numerical approach.The problem is firstly stated in its mechanical frame. Then the mathematical model is set up, emphasizing on the one hand the physical magnitude playing the part of the unknown function and on the other hand the laws of mechanics that lead to an ordinary differential equation or system. The solution is then obtained by specifying the mathematical methods described in the corresponding theoretical presentation. Finally a mechanical interpretation of the solution is provided, this giving rise to a complete knowledge of the studied phenomenon. The number of applications was increased, and many of these problems appear currently in engineering.

Table of Contents

Preface
Introduction
Generalities
Ordinary differential equations
Supplementary conditions associated to ODEs
The Cauchy (initial) problem
The two-point problem
Linear Odes of First and Second Order
Linear first order ODEs
Equations of the form
The linear homogeneous equation
The general case
The method of variation of parameters (Lagrange's method)
Differential polynomials
Linear second order ODEs
Homogeneous equations
Non-homogeneous equations. Lagrange's method
ODEs with constant coefficients
Order reduction
The Cauchy problem. Analytical methods to obtain the solution
Two-point problems (Picard)
Sturm-Liouville problems
Linear ODEs of special form
Applications
Linear Odes of Higher Order (N >2)
The general study of linear ODEs of order
Generalities
Linear homogeneous ODEs
The general solution of the non-homogeneous ODE
Order reduction
Linear ODEs with constant coefficients
The general solution of the homogeneous equation
The non-homogeneous ODE
Euler type ODEs
Fundamental solution. Green function
The fundamental solution
The Green function
The non-homogeneous problem
The homogeneous two-point problem. Eigenvalues
Applications
Linear Odss of First Order
The general study of linear first order ODSs
Generalities
The general solution of the homogeneous ODS
The general solution of the non-homogeneous ODS
Order reduction of homogeneous ODSs
Boundary value problems for ODSs
ODSs with constant coefficients
The general solution of the homogeneous ODS
Solutions in matrix form for linear ODSs with constant coefficients
Applications
Non-Linear Odes of First and Second Order
First order non-linear ODEs
Forms of first order ODEs and of their solutions
Geometric interpretation. The theorem of existence and uniqueness
Analytic methods for solving first order non-linear ODEs
First order ODEs integrable by quadratures
Non-linear second order ODEs
Cauchy problems
Two-point problems
Order reduction of second order ODEs
The Bernoulli-Euler equation
Elliptic integrals
Applications
Non-Linear Odss of First Order
Generalities
The general form of a first order ODS
The existence and uniqueness theorem for the solution of the Cauchy problem
The particle dynamics
First integrals of an ODS
Generalities
The theorem of conservation of the kinetic energy
The symmetric form of an ODS. Integral combinations
Jacobi's multiplier. The method of the last multiplier
Analytical methods of solving the Cauchy problem for non-linear ODSs
The method of successive approximations (Picard-Lindeloff)
The method of the Taylor series expansion
The linear equivalence method (LEM)
Applications
Variational Calculus
Necessary condition of extremum for functionals of integral type
Generalities
Functionals of the forma
Functionals of the forma
Functionals of integral type, depending on n functions
Conditional extrema
Isoperimetric problems
Lagrange's problem
Applications
Stability
Lyapunov Stability
Generalities
Lyapunov's theorem of stability
The stability of the solutions of dynamical systems
Autonomous dynamical systems
Long term behaviour of the solutions
Applications
Index
References
Table of Contents provided by Publisher. All Rights Reserved.

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