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CONTENTS xi

12.2. Second-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
12.2.1. Formulas for the General Solution. Some Transformations . . . . . . . . . . . . . . . 472
12.2.2. Representation of Solutions as a Series in the Independent Variable . . . . . . . . 475
12.2.3. Asymptotic Solutions .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 477
12.2.4. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
12.2.5. Eigenvalue Problems . .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 482
12.2.6. Theorems on Estimates and Zeros of Solutions .. .. .. ... .. .. .. .. .. .. .. .. 487
12.3. Second-Order Nonlinear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
12.3.1. Form of the General Solution. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 488
12.3.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.3.3. Methods of Regular Series Expansions with Respect to the Independent
Variable .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 492
12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations.
Painlev´ e Transcendents . .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 494
12.3.5. Perturbation Methods of Mechanics and Physics . . . . . . . . . . . . . . . . . . . . . . . 499
12.3.6. Galerkin Method and Its Modifications (Projection Methods) . . . . . . . . . . . . . 508
12.3.7. Iteration and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

12.4. Linear Equations of Arbitrary Order .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 514
12.4.1. Linear Equations with Constant Coefficients . . . . . . . ... .. .. .. .. .. .. .. .. 514
12.4.2. Linear Equations with Variable Coefficients .. .. .. .. ... .. .. .. .. .. .. .. .. 518
12.4.3. Asymptotic Solutions of Linear Equations . . . . . . . . . ... .. .. .. .. .. .. .. .. 522
12.4.4. Collocation Method and Its Convergence . . . . . . . . . . ... .. .. .. .. .. .. .. .. 523
12.5. Nonlinear Equations of Arbitrary Order .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 524
12.5.1. Structure of the General Solution. Cauchy Problem .. ... .. .. .. .. .. .. .. .. 524
12.5.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
12.6. Linear Systems of Ordinary Differential Equations . . . . . . . . . ... .. .. .. .. .. .. .. .. 528
12.6.1. Systems of Linear Constant-Coefficient Equations . . . ... .. .. .. .. .. .. .. .. 528
12.6.2. Systems of Linear Variable-Coefficient Equations . . . ... .. .. .. .. .. .. .. .. 539
12.7. Nonlinear Systems of Ordinary Differential Equations . . . . . . . ... .. .. .. .. .. .. .. .. 542
12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems . . . . . . . . . 542
12.7.2. Integrable Combinations. Autonomous Systems of Equations . . . . . . . . . . . . . 545
12.7.3. Elements of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
References for Chapter 12 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 550

13. First-Order Partial Differential Equations .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 553
13.1. Linear and Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
13.1.1. Characteristic System. General Solution . .. .. .. .. .. ... .. .. .. .. .. .. .. .. 553
13.1.2. Cauchy Problem. Existence and Uniqueness Theorem ... .. .. .. .. .. .. .. .. 556
13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . . 558
13.1.4. Quasilinear Equations of General Form. Generalized Solution, Jump
Condition, and Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
13.2. Nonlinear Equations .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 570
13.2.1. Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
13.2.2. Cauchy Problem. Existence and Uniqueness Theorem ... .. .. .. .. .. .. .. .. 576
13.2.3. Generalized Viscosity Solutions and Their Applications . . . . . . . . . . . . . . . . . 579
References for Chapter 13 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 584