# Differential Equations Topics

## 1. First Order Ordinary Differential Equations

- Introduction to Differential Equations
- Equilibrium Solutions to Differential Equations
- Direction Fields ( Examples 1 )
- Initial Value Problems
- The Method of Direct Integration ( Examples 1 | Examples 2 )
- The Method of Integrating Factors ( Examples 1 | Examples 2 | Examples 3 )
- Solving Separable Differential Equations ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
- Solving Differential Equations with Substitutions ( Examples 1 )
- Mixing and Dispersion with Differential Equations
- Compound Interest with Differential Equations
- The Existence/Uniqueness of Solutions to First Order Linear Differential Equations
- The Existence/Uniqueness of Solutions to General First Order Differential Equations
- Exact Differential Equations ( Examples 1 | Examples 2 )
- Bernoulli Differential Equations ( Examples 1 )
- Modelling Population Growth and Decay
- Stable, Semi-Stable, and Unstable Equilibrium Solutions ( Examples 1 )
- Phase Lines
- Euler's Method for Approximating Solutions to Differential Equations ( Examples 1 | Examples 2 | Examples 3 )
- The Method of Successive Approximations ( Examples 1 | Examples 2 )
- Summary of Techniques for Solving First Order Differential Equations

## 2. Second Order Differential Equations

- Second Order Homogenous Differential Equations
- Real and Distinct Roots of The Characteristic Equation
- The Existence/Uniqueness of Solutions to Second Order Linear Differential Equations
- The Principle of Superposition
- Wronskian Determinants of Two Functions
- Wronskian Determinants and Linear Homogenous Differential Equations
- Abel's Identity for Linear Homogenous Second Order Differential Equations
- Fundamental Solutions to Linear Homogenous Differential Equations
- Complex Roots of The Characteristic Equation ( Examples 1 )
- Repeated Roots of The Characteristic Equation ( Examples 1 )
- Reduction of Order on Second Order Linear Homogenous Differential Equations ( Examples 1 | Examples 2 )
- Euler Differential Equations ( Examples 1 )
- Second Order Nonhomogenous Differential Equations
- The Method of Undetermined Coefficients ( Examples 1 | Examples 2 )
- The Method of Variation of Parameters ( Examples 1 )
- Summary of Techniques for Solving Second Order Differential Equations

## 3. Higher Order Differential Equations

- The Existence/Uniqueness of Solutions to Higher Order Linear Differential Equations
- Higher Order Homogenous Differential Equations
- Wronskian Determinants of n Functions
- Wronskian Determinants and Higher Order Linear Homogenous Differential Equations
- Linear Independence/Dependence of a Set of Functions
- Higher Order Homogenous Differential Equations - Constant Coefficients
- Higher Order Homogenous Differential Equations - Real, Distinct Roots of The Characteristic Equation ( Examples 1 )
- Higher Order Homogenous Differential Equations - Complex Roots of The Characteristic Equation ( Examples 1 )
- Higher Order Homogenous Differential Equations - Repeated Roots of The Characteristic Equation ( Examples 1 )
- The Method of Undetermined Coefficients for Higher Order Nonhomogenous Differential Equations
- Differential Annihilators
- The Method of Annihilators ( Examples 1 )
- The Method of Variation of Parameters for Higher Order Nonhomogenous Differential Equations
- Summary of Techniques for Solving Higher Order Differential Equations

# 4. The Theory of First Order Ordinary Differential Equations

- First Order Ordinary Differential Equations
- Systems of First Order Ordinary Differential Equations
- nth Order Ordinary Differential Equations
- Converting nth Order ODEs to Systems of n First Order ODEs
- Preliminary Definitions for the Theory of First Order ODEs
- The Arzelà–Ascoli Theorem
- The Converse of the Arzelà–Ascoli Theorem
- Gronwall's Inequality
- Partially Order Sets and Zorn's Lemma
- ϵ-Approximate Solutions to Initial Values Problems of First Order ODEs
- The Local Existence Theorem for Solutions to Initial Value Problems of First Order ODEs
- Continuations of Solutions to First Order ODEs
- The Maximal Continuation Theorem of Solutions to First Order ODEs
- Continuations of Solutions to x' = h(t)g(x)
- The Lipschitz Uniqueness Theorem for Solutions to IVPs on First Order ODEs
- Banach's Fixed Point Theorem
- The Local Existence and Uniqueness Theorem via Banach's Fixed Point Theorem

- Vector and Matrix Norms
- Eigenvalues of Square Matrices
- Eigenvectors of Square Matrices
- Generalized Eigenvectors of Square Matrices
- The Jordan Canonical Form of a Square Matrix
- The Exponential of a Matrix

- Linear Homogeneous and Linear Nonhomogeneous Systems of First Order ODEs
- The Vector Space of the Solution Set to a Linear Homogeneous System of First Order ODEs
- Fundamental Sets of Solutions to a Linear Homogeneous System of First Order ODEs
- Fundamental Matrices to a Linear Homogeneous System of First Order ODEs
- Abel's Fundamental Matrix Formula
- The Determinant of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs
- Basic Properties of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs
- Criterion for a Matrix to be a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs
- The Change of Basis Theorems for Fundamental Matrices of a Linear Homogeneous System of First Order ODEs
- The State Transition Matrix to a Linear Homogeneous System of First Order ODEs
- Basic Properties of the State Transition Matrix to a Linear Homogeneous System of First Order ODEs
- The Unique Solutions to Linear Nonhomogeneous Systems of First Order ODEs

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###### References

- 1. Elementary Differential Equations and Boundary Value Problems (10th Edition) by William E. Boyce and Richard C. DiPrima.