Differential Equations

Reference

Reference 2

1. Introduction and Fundamentals
1.1 Definitions and Classification of Differential Equations (DEs)
1.2 Concept of a Solution to a Differential Equation

2. Techniques for Solving First-Order Differential Equations
2.1 Separable Equations
2.2 Exact Differential Equations
2.3 First-Order Linear Equations
2.4 Substitution-Based Methods
    2.4.1 Homogeneous Equations (Homogeneous Coefficients)
    2.4.2 Bernoulli Equations
    2.4.3 Other Useful Substitutions
2.5 Mixed First-Order Problems (Method Not Given in Advance)
2.6 Introduction to Computer-Aided Solutions for Differential Equations

3. Applications of First-Order Differential Equations
3.1 Decay and Growth Models
3.2 Newton’s Law of Cooling
3.3 Mixing Problems (Non-Reacting Fluids)
3.4 Basic Electric Circuit Models

4. Nth-Order Linear Differential Equations
4.1 Overview and Key Concepts
    4.1.1 Standard Form of an nth-Order Linear DE
    4.1.2 Differential Operator Notation
    4.1.3 Superposition Principle
    4.1.4 Linear Independence of Functions
4.2 Homogeneous Linear Differential Equations with Constant Coefficients
    4.2.1 General Solution of Homogeneous Linear ODEs
    4.2.2 Initial-Value and Boundary-Value Problems
4.3 Nonhomogeneous Linear Differential Equations with Constant Coefficients
    4.3.1 Structure of the General Solution
    4.3.2 Undetermined Coefficients Method
    4.3.3 Variation of Parameters Method
    4.3.4 Mixed Higher-Order Problems
4.4 Computer-Aided Solution Methods for Higher-Order Differential Equations

5. Laplace Transforms 

5.1 Definition of the Laplace Transform

5.2 Transforms of Elementary Functions 

5.3 The Second Shifting Theorem 

5.4 Derivative Properties of Transforms

5.5 Inverse Laplace Transforms

5.6 Using a Computer to Compute Laplace and Inverse Laplace Transforms

5.7 Laplace Transforms of Derivatives

5.8 Solving Initial Value Problems Using Laplace Transforms