Chapter 1 First-Order Ordinary Differential Equations

1-1 Introduction to Differential Equations

1-2 Separable First-Order ODEs

1-3 Exact ODEs and Integration Factor

1-4 Linear ODEs

1-5 Solving First-order ODEs with the Grouping Method

1-6 Application of First-Order ODEs



Chapter 2 High-Order Linear Ordinary Differential Equation

2-1 Basic Theories

2-2 Solving Higher-Order ODE with the Reduction of Order Method

2-3 Homogeneous Solutions of Higher-Order ODEs

2-4 Finding Particular Solution Using the Method of Undetermined Coefficients

2-5 Finding Particular Solution Using the Method of Variation of Parameters

2-6 Finding Particular Solution Using the Method of Inverse Differential Operators

2-7 Equidimensional Linear ODEs

2-8 The Applications of Higher-Order ODEs in Engineering



Chapter 3 Laplace Transform

3-1 The Definition of Laplace Transform

3-2 Basic Characteristic and Theorems

3-3 Laplace Transform of Special Functions

3-4 Laplace Inverse Transform

3-5 The Application of Laplace Transform



Chapter 4 Power Series Solution of Ordinary Differential Equations

4-1 Expansion at a Regular Point for Solving ODE

4-2 Regular Singular Point Expansion for Solving ODE (Selected Reading)



Chapter 5 Vector Operations and Vector Spaces

5-1 The Basic Operations of Vector

5-2 Vector Geometry

5-3 Vector Spaces Rn



Chapter 6 Matrix Operations and Linear Algebra

6-1 Matrix Definition and Basic Operations

6-2 Matrix Row (Column) Operations and Determinant

6-3 Solution to Systems of Linear Equations

6-4 Eigenvalues and Eigenvectors

6-5 Matrix Diagonalization

6-6 Matrix Functions



Chapter 7 Linear differential equation system

7-1 The Solution of a System of First-Order Simultaneous Linear Differential Equations

7-2 The Solution of a Homogeneous System of Simultaneous Differential Equations

7-3 Diagonalization of Matrix for Solving Non-Homogeneous System of Simultaneous Differential Equations



Chapter 8 Vector Function Analysis

8-1 Vector Functions and Differentiation

8-2 Directional Derivative

8-3 Line Integral

8-4 Multiple Integral

8-5 Surface Integral

8-6 Green’s Theorem

8-7 Gausss Divergence Theorem

8-8 Stokes’ Theorem



Chapter 9 Orthogonal Functions and Fourier Analysis

9-1 Orthogonal Functions

9-2 Fourier Series

9-3 Complex Fourier Series and Fourier Integral

9-4 Fourier Transform



Chapter 10 Partial Differential Equation

10-1 Introduction to Partial Differential Equation (PDE)

10-2 Solving Second-Order PDE Using the Method of Separation of Variables

10-3 Solving Non-Homogeneous Partial Differential Equation

10-4 Solving PDE Using Integral Transformations

10-5 Partial Differential Equations in Non-Cartesian Coordinate System



Chapter 11 Complex Analysis

11-1 Basic Concepts of Complex Number

11-2 Complex Functions

11-3 Differentiation of Complex Functions

11-4 Integration of Complex Functions

11-5 Taylor Series Expansion and Laurent Series Expansion

11-6 Residue Theorem

11-7 Definite Integral of Real Variable Functions