The proposed book is one of a series called "A Course of Higher Mathematics and Mathematical Physics" edited by A. N. Tikhonov, V. A. Ilyin and A. G. Sveshnikov. The book is based on a lecture course which, for a number of years now has been taught at the Physics Department and the Department of Computational Mathematics and Cybernetics of Moscow State University. The exposition reflects the present state of the theory of differential equations, as far as it is required by future specialists in physics and applied mathematics, and is at the same time elementary enough. An important part of the book is devoted to approximation methods for the solution and study of differential equations, e.g. numerical and asymptotic methods, which at the present time play an essential role in the study of mathematical models of physical phenomena. Less attention is paid to the integration of differential equations in elementary functions than to the study of algorithms on which numerical solution methods of differential equations for computers are based.
Table of ContentsI. Introduction.- § 1. The Concept of a Differential Equation.- § 2. Physical Problems Leading to Differential Equations.- II. General Theory.- § 1. Elementary Integration Methods.- § 2. Theorems on the Existence and Uniqueness of the Solution of the Initial Value Problem for a First Order Equation Resolved with Respect to the Derivative. The Euler Polygonal Line Algorithm.- § 3. Equations not Resolved with Respect to the Derivative.- § 4. Existence and Uniqueness Theorems for the Solution of Normal Systems.- § 5. Dependence of Solutions on Initial Values and Parameters.- § 6. The Method of Successive Approximations (Picard’s Method).- § 7. The Contraction Mapping Theorem.- III. Linear Differential Equations.- § 1. The Pendulum Equation as an Example of a Linear Equation. The Main Properties of Linear Equations with Constant Coefficients.- § 2. General Properties of n-th Order Equations.- § 3. Homogeneous n-th Order Linear Equations.- § 4. Non-homogeneous Linear n-th Order Equations.- § 5. Linear n-th Order Equations with Constant Coefficients.- § 6. Systems of Linear Equations. General Theory.- § 7. Systems of Linear Differential Equations with Constant Coefficients.- § 8. The Solutions in Power Series Form of Linear Equations.- IV. Boundary Value Problems.- § 1. Formulation of Boundary Value Problems and their Physical Meaning.- § 2. Non-homogeneous Boundary Value Problems.- § 3. Eigenvalue Problems.- V. Stability Theory.- § 1. Statement of the Problem.- § 2. Study of Stability in the First Approximation.- § 3. The Method of Lyapunov Functions.- § 4. The Study of Trajectories in a Neighbourhood of a Stationary Point.- VI. Numerical Methods for the Solution of Ordinary Differential Equations.- § 1. Numerical Methods for Solving Initial Value Problems.- § 2. Boundary Value Problems.- VII. Asymptotics of Solutions of Differential Equations with Respect to a Small Parameter.- § 1. Regular Perturbations.- § 2. Singular Perturbations.- VIII. First Order Partial Differential Equations.- § 1. Linear Equations.- § 2. Quasilinear Equations.