Contents
Preface
Chapter 1. First Order Differential Equations
1.1 Separable equations
1.2 Homogeneous equations
1.2.1 Quasihomogeneous Equations
1.3 Exact equations
1.3.1 Integrating Factors
1.4 Linear equations
1.4.1 Bernoullis Equation
1.4.2 Darboux's Equation
1.4.3 Riccati's Equation
1.4.4 Bool's Equation
1.5 Nonlinear equations
1.5.1 Solvable Equations. General Solution
1.5.2 Solvable Equations. Singular Solution
1.5.3 Unsolvable Equations
1.6 Applications in physics
1.6.1 Mechanics
1.6.2 Hydrodynamics
1.6.3 Electrical Networks
1.6.4 Kinetic Theory
1.6.5 Nuclear Physics
1.6.6 Optics
1.7 Miscellaneous problems
Chapter 2. N-th Order Differential Equations
2.1 Reduction of order
2.1.1 Simple Cases
2.1.2 Homogeneous Equations
2.1.3 Exact Equations
2.1.4 Linear Equations
2.1.5 The Initial Value Problem
2.2 Linear homogeneous equations
2.2.1 Exponential Solution
2.2.2 Power Solution
2.2.3 Transformations of Equation
2.2.4 The Initial Value Problem
2.3 Linear nonhomogeneous equations
2.3.1 Method of Variation of Parameters
2.3.2 Method of Undetermined Coefficients
2.3.3 The Influence Function
2.3.4 The Initial Value Problem
2.4 Linear equation with constant coefficients
2.4.1 The Homogeneous Equation with Constant Coefficients
2.4.2 The Complete Equation with Constant Coefficients. Method of Undetermined Coefficients
2.4.3 The Method of Variation of Parameters
2.4.4 Symbolic Methods
2.4.5 Laplace Transform
2.5 Equations with polynomial coefficients
2.5.1 Changes of Variable
2.5.2 Substitutions
2.5.3 Substitutions and Changes of Variable
Chapter 3. Linear Second Order Equations
3.1 Series solutions
3.1.1 Ordinary Point
3.1.2 Regular Singular Point
3.1.3 Irregular Singular Point
3.2 Linear boundary value problem
3.2.1 Homogeneous Problem
3.2.2 Nonhomogeneous Problem
3.2.3 Green's Function
3.3 Eigenvalues and eigenfunctions
3.3.1 Self-adjoint Problems
3.3.2 The Sturm-Liouville Problem
3.3.3 Nonhomogeneous Problem
Chapter 4. Systems Of Differential Equations
4.1 Linear systems with constant coefficients
4.1.1 Homogeneous Systems
4.1.2 Homogeneous Systems. Euler's Method
4.1.3 Euler's Method. Different Eigenvalues
4.1.4 Euler's Method. Repeated Eigenvalues
4.1.5 Repeated Eigenvalues. Method of Associated Vectors
4.1.6 Repeated Eigenvalues. Method of Undetermined Coefficients
4.1.7 Homogeneous Systems. Matrix Method
4.1.8 Nonhomogeneous Systems
4.1.9 Method of Variation of Parameters
4.1.10 Method of Undetermined Coefficients
4.1.11 Matrix Method
4.1.12 Initial Value Problem
4.1.13 Laplace Transform
4.1.14 Systems of Higher Order Equations
4.2 Linear systems
4.2.1 Solution by Eliminations
4.2.2 Matrix Method
4.2.3 Nonhomogeneous Linear Systems
4.2.4 Initial Value Problem
4.3 Nonlinear systems
4.3.1 Method of Eliminations
4.3.2 Method of Integrable Combinations
4.3.3 Systems of Bernoulli's Form
4.3.4 Method of Complex Variable
4.3.5 Systems of Canonical Form
Chapter 5. Partial Equations of the First Order
5.1 Linear partial equations
5.2 Pfaffian equation
5.2.1 Mayer's Method
5.3 Nonlinear partial equations
5.3.1 Lagrange Charpit's Method
Chapter 6. Nonlinear Equations And Stability
6.1 Phase plane. Linear systems
6.2 Almost linear systems
6.3 Liapunov's second method
Chapter 7. Calculus of Variations
7.1 Euler's equation
7.2 Conditional extremum
7.2.1 Isoperimetric Problem
7.3 Movable end points
7.4 Bolza problem
7.5 Euler-Poisson equation
7.6 Ostrogradsky equation
Chapter 8. Answers To Problems
A1.1 Separable equations
A1.2 Homogeneous equations
A1.3 Exact equations
A1.4 Linear equations
A1.5 Nonlinear equations
A1.6 Applications in physics
A1.7 Miscellaneous problems
A2.1 Reduction of order
A2.2 Linear homogeneous equations
A2.3 Linear nonhomogeneous equations
A2.4 Linear equation with constant coefficients
A2.5 Equations with polynomial coefficients
A3.1 Series solutions
A3.2 Linear boundary value problems
A3.3 Eigenvalues and eigenfunctions
A4.1 Systems with constant coefficients
A4.2 Linear systems
A4.3 Nonlinear systems
A5.1 Linear partial equations
A5.2 Pfaffian equation
A5.3 Nonlinear partial equations
A6.1 Phase plane. Linear systems
A6.2 Almost linear systems
A6.3 Liapunov's second method
A7.1 Euler's equation
A7.2 Conditional extremum
A7.2.1 Isoperimetric problem
A7.3 Movable end points
A7.4 Bolza problem
A7.5 Euler-Poisson equation
A7.6 Ostrogradsky equation
Bibliography
1-15
16-33
34-51
Index