Differential Equations and Linear Algebra

In this book, you will learn key techniques for solving differential equations that arise in science and engineering applications. You will also build a strong foundation in linear algebra concepts that support these methods.

Mathematics

Tác giả: Anup Lamichhane
Nhà xuất bản: Oho Northern University
Nguồn: LibreTexts
Giấy phép: CC BY-NC-SA
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 97 mục

01: First order ODEs
1.01: Introduction
1.02: Integrals as solutions
1.03: Slope fields
1.04: Separable Equations
1.05: Linear equations and the integrating factor
1.06: Substitution
1.07: Exact Equations
1.08: Autonomous equations
1.09: Existence and Uniqueness of Solutions of Nonlinear Equations
1.9E: Existence and Uniqueness of Solutions of Nonlinear Equations (Exercises)
1.10: Numerical methods- Eulers method
1.11: First Order Linear PDE
1.E: First order ODEs (Exercises)
02: Applications of First Order Equations
2.01: Growth and Decay
2.1E: Growth and Decay (Exercises)
2.02: Cooling and Mixing
2.2E: Cooling and Mixing (Exercises)
2.03: Elementary Mechanics
2.3E: Elementary Mechanics (Exercises)
2.04: Autonomous Second Order Equations
2.4E: Autonomous Second Order Equations (Exercises)
2.05: Applications to Curves
2.5E: Applications to Curves (Exercises)
03: Systems of Equations
3.01: Systems of Equations Geometry
3.02: Systems of Equations Algebraic Procedures
3.03: Gaussian Elimination
3.04: Uniqueness of the Reduced Row-Echelon Form
3.05: Applications of system of linear equations
3.06: Dimensionless Variables
3.07: Exercises
04: Matrices and Determinants
4.01: Matrix Arithmetic
4.02: The Transpose
4.03: The Identity and Inverses
4.04: Elementary Matrices
4.05: More on Matrix Inverses
4.06: LU
4.07: Determinants
4.08: Applications of the Determinant
4.09: Eigenvalues and Eigenvectors of a Matrix
4.10: Diagonalization
4.11: Determinant Exercises
4.E: Exercises
05: R
5.01: Vectors in R
5.02: Vector Algebra
5.03: Geometric Meaning of Vector Addition
5.04: Length of a Vector
5.05: Geometric Meaning of Scalar Multiplication
5.06: Spanning Linear Independence and Basis in R
5.07: Rank and Homogeneous Systems
5.E: Exercises
06: Higher order linear ODEs
6.01: Homogeneous Linear Equations
6.1.01: Homogeneous Linear Equations (Exercises)
6.02: Second order linear ODEs
6.03: Constant coefficient second order linear ODEs
6.04: Introduction to Linear Higher Order Equations
6.4.01: Higher order linear ODEs
6.4E: Introduction to Linear Higher Order Equations (Exercises)
6.05: Nonhomogeneous Linear Equations
6.5.01: Nonhomgeneous Linear Equations (Exercises)
6.06: The Method of Undetermined Coefficients I
6.6.01: The Method of Undetermined Coefficients I (Exercises)
6.07: The Method of Undetermined Coefficients II
6.7E: The Method of Undetermined Coefficients II (Exercises)
6.08: Reduction of Order
6.8E: Reduction of Order (Exercises)
6.09: Variation of Parameters
6.9E: Variation of Parameters (Exercises)
6.10: Mechanical Vibrations
6.11: Nonhomogeneous Equations
6.12: Forced Oscillations and Resonance
6.E: Higher order linear ODEs (Exercises)
07: The Laplace Transform
7.01: Introduction to the Laplace Transform
7.1E: Introduction to the Laplace Transform (Exercises)
7.02: The Inverse Laplace Transform
7.2E: The Inverse Laplace Transform (Exercises)
7.03: The Laplace Transform
7.04: Solution of Initial Value Problems
7.4E: Solution of Initial Value Problems (Exercises)
7.05: Transforms of derivatives and ODEs
7.06: Convolution
7.07: The Unit Step Function
7.7E: The Unit Step Function (Exercises)
7.08: Constant Coefficient Equations with Piecewise Continuous Forcing Functions
7.8.01: Constant Coefficient Equations with Piecewise Continuous Forcing Functions (Exercises)
7.09: Dirac Delta and Impulse Response
7.10: Constant Coefficient Equations with Impulses
7.10.01: Constant Coefficient Equations with Impulses (Exercises)
7.11: Solving PDEs with the Laplace Transform
7.12: A Brief Table of Laplace Transforms
7.E: The Laplace Transform (Exercises)