Differential Equations And The Calculus Of Variations | L. Elsgolts

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.

Contents

Chapter 1.

First-Order Differential Equations 19

1. First-Order Differential Equations Solved for the Derivative 19

2. Separable Equations 23

3. Equations That Lead to Separable Equations 29

4. Linear Equations of the First Order 32

5. Exact Differential Equations 37

6. Theorems of the Existence and Uniqueness of Solution of the equation dy/dx = f(x,y) 44

7. Approximate Me~hods of Integrating First-Order Equations 66

8. Elementary Types of Equations Not Solved for the Derivative 73

9. The Existence and Uniqueness Theorem for Differential Equations Not Solved for the Derivative. Singular Solutions 81

Problems 88

Chapter 2.

Differential Equations of the Second Order and Higher 91

1. The Existence and Uniqueness Theorem for an nth Order Differential Equation 91

2. The Most Elementary Cases of Reducing the Order 93

3. Linear Differential Equations of the nth Order 98

4. Homogeneous Linear Equations with Constant Coefficients and Euler's Equations 112

5. Nonhomogeneous Linear Equations 119

6. Nonhomogeneous Linear Equations with Constant Coefficients and Euler's Equations 130

7. Integration of Differential Equations by Means of Series 143

8. The Small Parameter Method and Its Application in the Theory of Quasilinear Oscillations 153

9. Boundary-Value Problems. Essentials

Problems

Chapter 3.

Systems of Differential Equations

1. Fundamentals

2. Integrating a System of Differential Equations by Reducing It to a Single Equation of Higher Order 179

3.Finding Integrable Combinations 186

4.Systems of Linear Differential Equations 189

5. Systems of Linear Differential Equations with Constant Coefficients 200

6. Approximate Methods of Integrating Systems of Differential Equations and Equations of Order n 206

Problems 209

Chapter 4.

Theory of Stability 211

1. Fundamentals 211

2. Elementary Types of Rest Points 214

3. Lyapunov's Second Method 223

4. Test for Stability Based on First Approximation 229

5. Criteria of Negativity of the Real Parts of All Roots of a Polynomial 236

6. The Case of a Small Coefficient of a Higher-Order Derivative 238

7. Stability Under Constantly Operating Perturbations 244

Problems 247

Chapter 5.

First-Order Partial Differential Equations 251

1. Fundamentals

2. Linear and Quasilinear First-Order Partial Differential Equations 253

3. Pfaffian Equations 265

4. First-Order Nonlinear Equations 271

Problems 288

PART TWO THE CALCULUS OF VARIATIONS 293

Chapter 6.

The Method of Variations in Problems with Fixed Boundaries 297

1. Variation and Its Properties 297

2. Euler's Equation 304

3. Functionals of the Form \int^{x_1} _{x_n} F (x,y_1, y_2,...y_n, y'_1, y'_2,...y'_n)dx 318

4. Functionals Dependent on Higher-Order Derivatives 321 5. Functionals Dependent on the Functions of Several Independent Variables 325

6. Variational Problems in Parametric Form 330

7. Some Applications 333

Problems 338

Chapter 7: Variational Problems with Moving Boundaries and Certain Other Problems 341

1. An Elementary Problem with Moving Boundaries 341

2. The Moving-Boundary Problem for a Functional.r,of the Form \int_{x_0}^{x_1}F(x, y, z, y', z') dx 347

3. Extremals with Corners 352

4. One-Sided Variations 360

Problems 363

Chapter 8.

Sufficient Conditions for an Extremum 365

1. Field of Extremals 365

2. The Function E (x, y, x', y') 371

3. Transforming the Euler Equations to the Canonical Form 383

Problems 387

Chapter 9.

Variational Problems Involving a Conditional Extremum 389

1. Constraints of the Form \phi (x, y_1, y_2, ., y_n) = 0 389

2. Constraints of the Form \phi (x, y_1, y_2, ., y_n, y'_1, y'_2,., y'_n) = 0 396

3. Isoperimetric Problems 399

Problems 407

Chapter 10.

Direct Methods In Variational Problems 408

1. Direct Methods 408

2. Euler's Finite-Difference Method 409

3. The Ritz Method 411

4. Kantorovich's Method 420

Problems 427

Answers to Problems 429

Recommended literature 436

Index 437