Differential Equations and the Calculus of Variations

Front Cover
Title Page
Contents
PART ONE Differential Equations
	Introduction
	1. First-order differential equations
		1. First-Order Differential EquationsSolved for the Derivative
		2. Separable Equations
		3. Equations That Lead to Separable Equations
		4. Linear Equations of the First Order
		5. Exact Differential Equations
		6. Theorems of the Existence and Uniqueness of Solution of the equation dy/dx = f(x,y)
		7. Approximate Methods of Integrating First-Order Equations
		8. Elementary Types of Equations Not Solved for the Derivative
		9. The Existence and Uniqueness Theoremfor Differential Equations Not Solvedlor the Derivative. Singular Solutions
		PROBLEMS ON CHAPTER I
	2. Differential equations of the secondorder and higher
		1. The Existence and Uniqueness Theorem for an nth OrderDifferential Equation
		2. The Most Elementary Cases of Reducing the Order
		3. Linear Differential Equations of the nth Order
		4. Homogeneous Linear Equations with Constant Coefficients and Euler\'s Equations
		5. Nonhomogeneous linear Equations
		6. Nonhomogeneous Linear Equations with Constant Coefficients and Euler\'s Equations
		7. Integration of Differential Equations by Means of Series
		8. The Small Parameter Method and Its Application in the Theory of Quasilinear Oscillations
		9. Boundary-Value Problems. Essentials
		PROBLEMS ON CHAPTER 2
	3. Systems of differential equations
		1. Fundamentals
		2. Integrating a System of Differential Equations by ReducingIt to a Single Equation of Higher Order
		3. Finding Integrable Combinations
		4. Systems of Linear Differential Equations
		5. Systems of Linear Differential Equations with Constant Coefficients
		6. Approximate Methods of Integrating Systemsof Differential Equations and Equations of Order n
		PROBLEMS ON CHAPTER 3
	4. Theory of stability
		1. Fundamentals
		2. Elementary Types of Rest Points
		3. Lyapunov\'s Second Method
		4. Test for Stability Based on First Approximation
		5. Criteria of Negativity of the Real Parts of All Rootsof a Polynomial
		6. The Case of a Small Coefficient of a Higher-Order Derivative
		PROBLEMS ON CHAPTER 4
	5. First-order partial differential equations
		1. Fundamentals
		2. Linear and Quasi linear First-Order Partial Differential Equations
		3. Pfaffian Equations
		4. First-Order Nonlinear Equations
		PROBLEMS ON CHAPTER 5
PART TWO The Calculus of Variations
	Introduction
	6. The method of variations in problemswith fixed boundaries
		1. Variation and Its Properties
		2. Euler\'s Equation
		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
		4. Functionals Dependent on Higher-Order Derivatives
		5. Functlonals Dependent on the Functionsof Several Independent Variables
		6. Variational Problems in Parametric Form
		7. Some Applications
		PROBLEMS ON CHAPTER 6
	7. Variational problems with moving boundaries and certain other problems
		1. An Elementary Problem with Moving Boundaries
		2. The Moving-Boundary Problem for a Functional.r,of the Form \\int_{x_0}^{x_1}F(x, y, z, y\', z\') dx
		3. Extremals with Corners
		4. One-Sided Variations
		PROBLEMS ON CHAPTER 7
	8. Sufficient conditions for an extremum
		1. Field of Extremals
		2. The Function E(x, y, p, y\')
		3. Transforming the Euler Equations to the Canonical Form
		PROBLEMS ON CHAPTER 8
	9. Variational problemsinvolving a conditional extremum
		1. Constraints of the Form \\phi (x, y_1, y_2, ... , y_n) = 0
		2. Constraints of the Form \\phi (x, y_1, y_2, ., y_n, y\'_1, y\'_2,., y\'_n) = 0
		3. lsoperimetrlc Problems
		PROBLEMS ON CHAPTER 9
	10. Direct methods in variationalproblems
		1. Direct Methods
		2. Euler\'s Finite-Difference Method
		3. The Ritz Method
		4. Kantorovlch\'s Method
		PROBLEMS ON CHAPTER 10
Answers to problems
Recommended Literature
Index