Ordinary Differential Equations

A Graduate Text

ORDINARY DIFFERENTIAL EQUATIONS: A Graduate Text presents a systematic and comprehensive introduction to ODEs for graduate and postgraduate students. The systematic organized text on differential inequalities, Gronwall's inequality, Nagumo's theorems, Osgood's criteria and applications of different equations of first order is dealt with in a greater depth. Les mer
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ORDINARY DIFFERENTIAL EQUATIONS: A Graduate Text presents a systematic and comprehensive introduction to ODEs for graduate and postgraduate students. The systematic organized text on differential inequalities, Gronwall's inequality, Nagumo's theorems, Osgood's criteria and applications of different equations of first order is dealt with in a greater depth. The book discusses qualitative and quantitative aspects of the Strum - Liouville problems, Green's function, integral equations, Laplace transform and is supported by a number of worked-out examples in each lesson to make the concepts clear. A lot of stress on stability theory is laid down, especially on Lyapunov and Poincare stability theory. A numerous figures in various lessons (in particular lessons dealing with stability theory) have been added to clarify the key concepts in DE theory. Nonlinear oscillation in conservative systems and Hamiltonian systems highlights basic nature of the systems considered. Perturbation techniques lesson deals in fairly details to understand basic nature and approximate solutions of nonlinear DEs like: Rayleigh equation, Duffing equation, Lienard equation, van der Pol equation.
KEY FEATURES: Contains over 596 worked out examples and over 569 unsolved (many with answers or hints) in the book Good bibliography and appendices Can be used as a reference after the students have completed learning elementary part of the subject

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Innholdsfortegnelse

Preface / Preliminaries / Elementary Odefo / Nonlinear Odefos / Solution in Series for Odefo / Existence and Uniqueness Theorems / Successive Approximations / Differential Inequalities and Maximal - Minimal Solutions / The Gronwall's Integral Inequality and Nagumo's Criteria / Osgood's Uniqueness Theorem / The Solution that Varies Continuously with a Parameter / Applications of First Order and First Degree Des / Genesis of Odesos / Solutions of Linear Odeso / Solution of BVPs and IVPs / Sturm-Liouville Equations / Oscillations of Solutions / Prufer Substitution / Orthogonal Functions / Green's Function / Modified Green's Function / Laplace Transform / Linear Integral Equations / Separable Kernels in Integral Equations / Hilbert Schmidt Theory / Matrices and Linear Systems / System of First Order Linear Equations / Non-Autonomous Systems / Stability in Nonlinear Systems / Linearization Methods for Autonomous Systems / Lyapunov's Functions / Lyapunov Analysis of Nonlinear Time Varying Systems / Extensions of Lyapunov Method / Limit Cycles / Limit Sets and Trajectories / Periodic Solutions / The Stability of Periodic Solutions / Competing Species / Poincare Stability Theory / Stability of Quasi-Linear Systems / Poincare Index and Poincare Map / Elementary Bifurcation Theory / The Conservative Systems / Special Types Nonlinear Equations / Duffing Equation / Perturbation Methods / Appendices / Bibliography / Index.