A.N. Tikhonov, A.B. Vasileva, A.G. Sveshnikov ; translated from the Russian by A.B. Sossinskij
Translated from the Russian by Sossinskij, A.B.
I. Introduction.- x 1. The Concept of a Differential Equation.- x 2. Physical Problems Leading to Differential Equations.- II. General Theory.- x 1. Elementary Integration Methods.- x 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.- x 3. Equations not Resolved with Respect to the Derivative.- x 4. Existence and Uniqueness Theorems for the Solution of Normal Systems.- x 5. Dependence of Solutions on Initial Values and Parameters.- x 6. The Method of Successive Approximations (Picard's Method).- x 7. The Contraction Mapping Theorem.- III. Linear Differential Equations.- x 1. The Pendulum Equation as an Example of a Linear Equation. The Main Properties of Linear Equations with Constant Coefficients.- x 2. General Properties of n-th Order Equations.- x 3. Homogeneous n-th Order Linear Equations.- x 4. Non-homogeneous Linear n-th Order Equations.- x 5. Linear n-th Order Equations with Constant Coefficients.- x 6. Systems of Linear Equations. General Theory.- x 7. Systems of Linear Differential Equations with Constant Coefficients.- x 8. The Solutions in Power Series Form of Linear Equations.- IV. Boundary Value Problems.- x 1. Formulation of Boundary Value Problems and their Physical Meaning.- x 2. Non-homogeneous Boundary Value Problems.- x 3. Eigenvalue Problems.- V. Stability Theory.- x 1. Statement of the Problem.- x 2. Study of Stability in the First Approximation.- x 3. The Method of Lyapunov Functions.- x 4. The Study of Trajectories in a Neighbourhood of a Stationary Point.- VI. Numerical Methods for the Solution of Ordinary Differential Equations.- x 1. Numerical Methods for Solving Initial Value Problems.- x 2. Boundary Value Problems.- VII. Asymptotics of Solutions of Differential Equations with Respect to a Small Parameter.- x 1. Regular Perturbations.- x 2. Singular Perturbations.- VIII. First Order Partial Differential Equations.- x 1. Linear Equations.- x 2. Quasilinear Equations.