Part 1 Ordinary Differential Equations

Chapter 1 Introduction to Differential Equations
1.1 Definitions and Terms
1.2 Initial value problem
1.3 Differential equations as mathematical models
Chapter 1 Review Questions

Chapter 2 First-Order Differential Equations
2.1 Drawing a solution curve without a solution
2.2 Variable separation
2.3 Linear Differential Equations
2.4 Exact differential equations
2.5 Substitution method
2.6 Numerical solution
2.7 Linear Model
2.8 Nonlinear models
2.9 Modeling using simultaneous first-order differential equations
Chapter 2 Review Questions

Chapter 3 Higher-Order Differential Equations
3.1 Linear Differential Equation Theory
3.2 Lowering the coefficient
3.3 Linear differential equations with constant coefficients
3.4 Undetermined coefficient method
3.5 Parameter variation method
3.6 Cauchy-Euler equation
3.7 Nonlinear equations
3.8 Linear Models: Initial Value Problems
3.9 Linear Models: The Boundary Problem
3.10 Green's function
3.11 Nonlinear models
3.12 Solution of simultaneous linear differential equations
Chapter 3 Review Questions

Chapter 4 Laplace Transforms
4.1 Definition of Laplace transform
4.2 Inverse transformation and derivative transformation
4.3 Parallel Translation Theorem
4.4 Additional operational properties
4.5 Dirac delta function
4.6 Systems of linear differential equations
Chapter 4 Review Questions

Chapter 5 Series Solutions to Linear Differential Equations
5.1 Solution for the common point
5.2 Solution to singularity
5.3 Special functions
Chapter 5 Review Questions

Chapter 6 Numerical Solutions of Ordinary Differential Equations
6.1 Euler's method and error analysis
6.2 Runge-Kutta method
6.3 Multi-step method
6.4 Higher-order equations and simultaneous equations
6.5 Second-order boundary value problem
Chapter 6 Review Questions


Part 2 Vectors, Matrices, and Vector Calculus

Chapter 7 Vectors
7.1 Vectors in two-dimensional coordinate space
7.2 Vectors in three-dimensional coordinate space
7.3 Inner product
7.4 Vector product
7.5 Lines and planes in three-dimensional coordinate space
7.6 Vector Space
7.7 Gram-Schmidt orthogonalization process
Chapter 7 Review Questions

Chapter 8 Matrices
8.1 Matrix Algebra
8.2 Systems of linear algebraic equations
8.3 Matrix coefficients
8.4 Determinant
8.5 Properties of determinants
8.6 Inverse matrix
8.7 Cramer's Law
8.8 Eigenvalue Problem
8.9 Powers of matrices
8.10 Orthogonal matrices
8.11 Approximation of Eigenvalues
8.12 Diagonalization
8.14 LU factorization
8.14 Cryptography
8.15 Error Correction Code
8.16 Least Squares Method
8.17 Discrete Compartment Model
Chapter 8 Review Questions

Chapter 9 Vector Calculus
9.1 Vector functions
9.2 Curvilinear motion
9.3 Curvature
9.4 Partial derivatives
9.5 Directional derivatives
9.6 Tangent plane and normal
9.7 Rotation and Divergence
9.8 Line integral
9.9 Path Independence of Line Integrals
9.10 Double Integral
9.11 Double integral in polar coordinates
9.12 Green's Theorem
9.13 Area
9.14 Stokes' theorem
9.15 Triple Integral
9.16 Divergence Theorem
9.17 Variable transformation of double integral
Chapter 9 Review Questions


Part 3: Systems of Differential Equations

Chapter 10 Linear Differential Semiformulas
10.1 Basic Concepts
10.2 Preliminary linear system
10.3 Solution by diagonalization
10.4 Non-linear system
10.5 Matrix Exponential Function
Chapter 10 Review Questions

Chapter 11: Systems of Nonlinear Differential Equations
11.1 Autonomous system
11.2 Stability of linear systems
11.3 Linearization and local stability
11.4 Autonomous Systems as Mathematical Models
11.5 Periodicity, Extreme Circulation, and Global Stability
Chapter 11 Review Questions

Appendix
Appendix A.
Function defined through integration
Appendix B.
Derivative and integral formulas
Appendix C.
Laplace transformation table
Appendix D.
isometric projections
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