This text is meant for students of higher schools and deals with the most important sections of mathematics-differential equations and the calculus of variations. The book contains a large number of examples and problems with solutions involving applications of mathematics to physics and mechanics.The book was translated from the Russian by George Yankovsky and was first published by Mir Publishers in 1970. There were two reprints one in 1973 and one in 1977. The book here is to the third reprint.ContentsChapter 1.First-Order Differential Equations 191. First-Order Differential Equations Solved for the Derivative 192. Separable Equations 233. Equations That Lead to Separable Equations 294. Linear Equations of the First Order 325. Exact Differential Equations 376. Theorems of the Existence and Uniqueness of Solution of the equation dy/dx = f(x,y) 447. Approximate Me~hods of Integrating First-Order Equations 668. Elementary Types of Equations Not Solved for the Derivative 739. The Existence and Uniqueness Theorem for Differential Equations Not Solved for the Derivative. Singular Solutions 81Problems 88Chapter 2.Differential Equations of the Second Order and Higher 911. The Existence and Uniqueness Theorem for an nth Order Differential Equation 912. The Most Elementary Cases of Reducing the Order 933. Linear Differential Equations of the nth Order 984. Homogeneous Linear Equations with Constant Coefficients and Euler's Equations 1125. Nonhomogeneous Linear Equations 1196. Nonhomogeneous Linear Equations with Constant Coefficients and Euler's Equations 1307. Integration of Differential Equations by Means of Series 1438. The Small Parameter Method and Its Application in the Theory of Quasilinear Oscillations 1539. Boundary-Value Problems. EssentialsProblemsChapter 3.Systems of Differential Equations1. Fundamentals2. Integrating a System of Differential Equations by Reducing It to a Single Equation of Higher Order 1793.Finding Integrable Combinations 1864.Systems of Linear Differential Equations 1895. Systems of Linear Differential Equations with Constant Coefficients 2006. Approximate Methods of Integrating Systems of Differential Equations and Equations of Order n 206Problems 209Chapter 4.Theory of Stability 2111. Fundamentals 2112. Elementary Types of Rest Points 2143. Lyapunov's Second Method 2234. Test for Stability Based on First Approximation 2295. Criteria of Negativity of the Real Parts of All Roots of a Polynomial 2366. The Case of a Small Coefficient of a Higher-Order Derivative 2387. Stability Under Constantly Operating Perturbations 244Problems 247Chapter 5.First-Order Partial Differential Equations 2511. Fundamentals2. Linear and Quasilinear First-Order Partial Differential Equations 2533. Pfaffian Equations 2654. First-Order Nonlinear Equations 271Problems 288PART TWO THE CALCULUS OF VARIATIONS 293Chapter 6.The Method of Variations in Problems with Fixed Boundaries 2971. Variation and Its Properties 2972. Euler's Equation 3043. Functionals of the Form \int^{x_1} _{x_n} F (x,y_1, y_2,...y_n, y'_1, y'_2,...y'_n)dx 3184. Functionals Dependent on Higher-Order Derivatives 321 5. Functionals Dependent on the Functions of Several Independent Variables 3256. Variational Problems in Parametric Form 3307. Some Applications 333Problems 338Chapter 7: Variational Problems with Moving Boundaries and Certain Other Problems 3411. An Elementary Problem with Moving Boundaries 3412. The Moving-Boundary Problem for a Functional.r,of the Form \int_{x_0}^{x_1}F(x, y, z, y', z') dx 3473. Extremals with Corners 3524. One-Sided Variations 360Problems 363Chapter 8.Sufficient Conditions for an Extremum 3651. Field of Extremals 3652. The Function E (x, y, x', y') 3713. Transforming the Euler Equations to the Canonical Form 383Problems 387Chapter 9.Variational Problems Involving a Conditional Extremum 3891. Constraints of the Form \phi (x, y_1, y_2, ., y_n) = 0 3892. Constraints of the Form \phi (x, y_1, y_2, ., y_n, y'_1, y'_2,., y'_n) = 0 3963. Isoperimetric Problems 399Problems 407Chapter 10.Direct Methods In Variational Problems 4081. Direct Methods 4082. Euler's Finite-Difference Method 4093. The Ritz Method 4114. Kantorovich's Method 420Problems 427Answers to Problems 429Recommended literature 436Index 437