This volume studies differential equations involving functions of a single variable.
This volume studies differential equations involving functions of a single variable. It develops existence theory, qualitative behavior, and analytical and numerical methods.
Part I. First-Order Equations
Chapter 1. Basic Concepts
1.1 Definitions and examples 1.2 Solutions and integral curves 1.3 Initial value problems 1.4 Direction fields 1.5 Modeling examples
Chapter 2. Separable and Exact Equations
2.1 Separable equations 2.2 Exact equations 2.3 Integrating factors 2.4 Applications 2.5 Examples
Chapter 3. Linear First-Order Equations
3.1 General form 3.2 Integrating factor method 3.3 Existence and uniqueness 3.4 Applications 3.5 Examples
Part II. Higher-Order Linear Equations
Chapter 4. Linear Differential Equations
4.1 n-th order equations 4.2 Homogeneous solutions 4.3 Superposition principle 4.4 Wronskian 4.5 Examples
Chapter 5. Nonhomogeneous Equations
5.1 Particular solutions 5.2 Method of undetermined coefficients 5.3 Variation of parameters 5.4 Applications 5.5 Examples
Chapter 6. Constant Coefficient Equations
6.1 Characteristic equation 6.2 Real and complex roots 6.3 Repeated roots 6.4 Applications 6.5 Examples
Part III. Systems of Differential Equations
Chapter 7. Linear Systems
7.1 Matrix formulation 7.2 Fundamental matrix 7.3 Eigenvalue methods 7.4 Stability analysis 7.5 Examples
Chapter 8. Nonlinear Systems
8.1 Phase space 8.2 Equilibrium points 8.3 Linearization 8.4 Stability types 8.5 Examples
Chapter 9. Autonomous Systems
9.1 Definition 9.2 Phase portraits 9.3 Limit cycles 9.4 Applications 9.5 Examples
Part IV. Qualitative Theory
Chapter 10. Existence and Uniqueness
10.1 Picard–Lindelöf theorem 10.2 Continuation of solutions 10.3 Dependence on initial conditions 10.4 Applications 10.5 Examples
Chapter 11. Stability Theory
11.1 Lyapunov stability 11.2 Asymptotic stability 11.3 Lyapunov functions 11.4 Applications 11.5 Examples
Chapter 12. Oscillation and Periodicity
12.1 Oscillatory solutions 12.2 Sturm comparison 12.3 Periodic solutions 12.4 Applications 12.5 Examples
Part V. Special Methods
Chapter 13. Series Solutions
13.1 Power series methods 13.2 Frobenius method 13.3 Regular singular points 13.4 Applications 13.5 Examples
Chapter 14. Transform Methods
14.1 Laplace transform 14.2 Initial value problems 14.3 Convolution 14.4 Applications 14.5 Examples
Chapter 15. Perturbation Methods
15.1 Regular perturbation 15.2 Singular perturbation 15.3 Multiple scales 15.4 Applications 15.5 Examples
Part VI. Boundary Value Problems
Chapter 16. Boundary Conditions
16.1 Types of boundary conditions 16.2 Sturm–Liouville problems 16.3 Eigenvalue problems 16.4 Applications 16.5 Examples
Chapter 17. Green’s Functions
17.1 Construction 17.2 Properties 17.3 Applications 17.4 Examples 17.5 Connections
Chapter 18. Variational Methods
18.1 Euler–Lagrange equations 18.2 Functional minimization 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Numerical Methods
Chapter 19. Initial Value Problems
19.1 Euler method 19.2 Runge–Kutta methods 19.3 Stability 19.4 Error analysis 19.5 Applications
Chapter 20. Boundary Value Problems
20.1 Finite difference methods 20.2 Shooting method 20.3 Stability and convergence 20.4 Applications 20.5 Examples
Chapter 21. Computational Tools
21.1 Software systems 21.2 Symbolic methods 21.3 Numerical libraries 21.4 Visualization 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Nonlinear dynamics 22.2 Chaos (overview) 22.3 Delay differential equations 22.4 Functional differential equations 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Stability challenges 23.2 Nonlinear classification 23.3 Computational complexity 23.4 Analytical difficulties 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of ODE theory 24.2 Key contributors 24.3 Evolution of methods 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common solution techniques B. Stability criteria summary C. Proof techniques checklist D. Numerical method tables E. Cross-reference to other MSC branches
This volume develops ordinary differential equations as a central tool for modeling and analysis. It emphasizes both exact solutions and qualitative behavior.