ORDINARY DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS

Transcription

1 ORDINARY DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS Book of Problems M. V. Makarets Kiev T. Shevchenko University, Ukraine V. Yu. Reshetnyak Institute of Surface Chemistry, Ukraine.0 World Scientific! Singapore»New Jersey London Hong Kong

2 Contents PREFACE ix 1 FIRST ORDER DIFFERENTIAL EQUATIONS Separable equations Homogeneous equations Quasihomogeneous Equations Exact equations Integrating Factors Linear equations Bernoulli's Equation Darboux's Equation Riccati's Equation Bool's Equation Nonlinear equations Solvable Equations. General Solution Solvable Equations. Singular Solution Unsolvable Equations Applications in physics Mechanics Hydrodynamics Electrical Networks Kinetic Theory Nuclear Physics Optics Miscellaneous problems 74 2 N-th ORDER DIFFERENTIAL EQUATIONS Reduction of order Simple Gases Homogeneous Equations Exact Equations Linear Equations The Initial Value Problem Linear homogeneous equations 87 v

3 vi CONTENTS Exponential Solution Power Solution Transformations of Equation The Initial Value Problem Linear nonhomogeneous equations Method of Variation of Parameters Method of Undetermined Coefficients The Influence Function The Initial Value Problem Linear equation with constant coefficients The Homogeneous Equation with Constant Coefficients The Complete Equation with Constant Coefficients. Method of Undetermined Coefficients The Method of Variation of Parameters Symbolic Methods Laplace Transform Equations with polynomial coefficients Changes of Variable Substitutions Substitutionsand Changes of Variable Series Solutions LINEAR SECOND ORDER EQUATIONS Series Solutions Ordinary Point Regulär Singular Point Irregulär Singular Point Linear boundary value problem Homogeneous Problem Nonhomogeneous Problem Green's Function Eigenvalues and eigenfunctions Self-adjoint Problems The Sturm-Liouville Problem Nonhomogeneous Problem SYSTEMS OF DIFFERENTIAL EQUATIONS Linear Systems with constant coefficients Homogeneous Systems Homogeneous Systems. Euler's Method Euler's Method. Different Eigenvalues Euler's Method. Repeated Eigenvalues Repeated Eigenvalues. Method of Associated Vectors 194

4 CONTENTS vii Repeated Eigenvalues. Method of Undetermined Coefficients Homogeneous Systems. Matrix Method Nonhomogeneous Systems Method of Variation of Parameters Method of Undetermined Coefficients Matrix Method Initial Value Problem Laplace Transform Systems of Higher Order Equations Linear Systems Solution by Eliminations Matrix Method Nonhomogeneous Linear Systems Initial Value Problem Nonlinear Systems Method of Eliminations Method of Integrable Combinations Systems of Bernoulli's Form Method of Complex Variable Systems of Canonical Form PARTIAL EQUATIONS OF THE FIRST ORDER Linear partial equations Pfaffian equation Mayer's Method Nonlinear partial equations Lagrange - Charpit's Method NONLINEAR EQUATIONS AND STABILITY Phase plane. Linear Systems Almost linear Systems Liapunov's second method CALCULUS OF VARIATIONS Euler's equation Conditional extremum Isoperimetric Problem Movable end points Bolza problem Euler-Poisson equation Ostrogradsky equation 303

5 viii CONTENTS 8 ANSWERS TO PROBLEMS Separable equations Homogeneous equations Exact equations Linear equations Nonlinear equations Applications in physics Miscellaneous problems Reduction of order Linear homogeneous equations Linear nonhomogeneous equations Linear equation with constant coefficients Equations with polynomial coefficients Series Solutions Linear boundary value problems Eigenvalues and eigenfunctions Systems with constant coefficients Linear Systems Nonlinear Systems Linear partial equations Pfaffian equation Nonlinear partial equations Phase plane. Linear Systems Almost linear Systems Liapunov's second method Euler's equation Conditional extremum Isoperimetric problem Movableend points Bolza problem Euler-Poisson equation Ostrogradsky equation 363 BIBLIOGRAPHY 365 INDEX 369