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| Differential Equations and calculus of variations book by L. Elsgolts-signed |
COURSE CONTENTS:
PART
ONE
DIFFERENTIAL
EQUATIONS
Introduction
Chapter 1. First-Order
Dillerential Equations
1. First-Order Differential Equations
Solved 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 The uniqueness of Solution of the Equation
7. Approximate Me~hods
of Integrating First-Order·
Equations
8. Elementary Types of Equations Not
Solved for the Derivative
9. The Existence and Uniqueness Theorem
for Differential Equations Not Solved for the Derivative. Singular Solutions Problems
Chapter 2. Differential
Equations of the Second Order and Higher
1. The Existence and Uniqueness
Theorem for an ntn Order Differential 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 utilizing Series
8. The Small Parameter Method and its
Application in the Theory of Quasilinear Oscillations
9.
Boundary-Value Problems. Essentials Problems
Chapter 3. Systems of
Differential Equations
I. Fundamentals
2. Integrating a System of Differential
Equations by Reducing It 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
Systems of Differential Equations and Equations of Order n.
Problems
Chapter 4. Theory of
Stability
I. 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 Roots of a Polynomial
6. The Case of a Small Coefficient of a
Higher-Order Derivative
7. Stability Under Constantly Operating
Perturbations
Problems
Chapter 5. First-Order
Partial Differential Equations
I. Fundamental
2. Linear and Quasilinear First-Order
Partial Differential Equations
3. Pfaffian Equations
4. First-Order Nonlinear Equations
Problems
PART TWO
THE
CALCULUS OF VARIATIONS
Introduction
Chapter 6. The Method of
Variations in Problems with Fixed Boundaries
I. Variation
and Its Properties
2. Euler's Equation
S. Functionals of the Form
4. Functionals Dependent on"•
Higher-Order Derivatives
5. Functionals Dependent on the
Functions of Several Independent Variables
6. Variational Problems in Parametric
Form
7. Some Applications
Problem
Chapter 7: Variational
Problems with Moving Boundaries and Certain Other Problems
1. An Efernentary Problem with Moving
Boundaries
2. The Moving-Boundary Problem for a
Functional of the Form
3. Extremals with Corners
4. One-Sided Variations Problems
Chapter 8. Sufficient
Conditions for an Extremum
I. Field
of Extremals
2. Function E (x, g, p, g')
3. Transforming the Euler Equations to
the Canonical Form
Problems
Chapter 9. Variational
Problems Involving a Conditional Extremum
1. Constraints of the Form
2. Constraints of the Form
3. Isoperimetric Problems
Problems
Chapter 10. Direct
Methods in Variational Problems
1. Direct Methods
2. Euler's Finite-Difference Method
3. The Ritz Method
4. Kantorovich's Method
Problems
Answers to Problems
Recommended literature
Index
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