Odinary Differential Equations

Basics

• Definition and Examples of Differential Equations
• Classification of Differential Equations

First-Order Differential Equations

Linear Differential Equations

• Separable Differential Equations
• Definition and Criterion of Exact Differential Equations
  • Solution of Exact Differential Equations
• Method of Integrating Factors
• Homogeneous Functions and Differential Equations
• First-Order Linear Differential Equations with Constant Coefficients
• Euler Differential Equations
  • Nonhomogeneous Euler Differential Equations
• Picard’s Method
• Existence and Uniqueness of Solutions to Initial Value Problems for First-Order Ordinary Differential Equations

Nonlinear Differential Equations

• Bernoulli Differential Equations
• Riccati Differential Equations
• Clairaut’s Differential Equations

Second-Order Linear Differential Equations

Homogeneous Equations

• Meaning of Homogeneity in Differential Equations
• Second-Order Linear Homogeneous Differential Equations with Constant Coefficients and Characteristic Equation
• Linear Superposition of Solutions for Linear Homogeneous Differential Equations
• Fundamental Set of Solutions and Wronskian for Second-Order Linear Homogeneous Differential Equations
• Wronskian of Two Solutions of a Second-Order Linear Differential Equation
• Solving Second-Order Linear Homogeneous Differential Equations
• Method to Find the Second Solution of a Second-Order Differential Equation

Nonhomogeneous Equations

• General Solution of Second-Order Linear Nonhomogeneous Differential Equations

Series Solutions of Second-Order Linear Differential Equations

• Solving Differential Equations Using Series Solutions
• Frobenius Method
• Series Solutions of Bessel’s Differential Equation: Bessel Functions of the First Kind
• Second Series Solution of Bessel’s Differential Equation: Bessel Functions of the Second Kind, Neumann Functions, Weber Functions
• Differential Equations with Bessel Functions as Solutions
• Series Solutions of Chebyshev’s Differential Equation: Chebyshev Polynomials
• Chebyshev’s Differential Equation and Chebyshev Polynomials
• Series Solutions of Legendre’s Differential Equation: Legendre Polynomials
• Series Solutions of the Associated Legendre Differential Equation
• Legendre Differential Equation in Trigonometric Form
• Series Solutions of Hermite’s Differential Equation: Hermite Polynomials
• Series Solutions of Laguerre’s Differential Equation
• Series Solutions of the Airy Differential Equation

Laplace Transforms

• Definition and Existence of Laplace Transforms
• Linearity of Laplace Transforms
• Solving Second-Order Linear Nonhomogeneous Differential Equations Using Laplace Transforms
• Convolution in Laplace Transforms

Laplace Transforms of Various Functions

• Laplace Transform Table
• Constant Functions
• Exponential Functions
• Trigonometric Functions
• Hyperbolic Functions
• Polynomial Functions
• First-Order Derivatives
• n-th Order Derivatives
• Step Functions
• $\cal{L}\left\{ e^{ct}f(t) \right\}$: Translation of Laplace Transforms
• $\cal{L}\left\{ f(ct) \right\}$: Dilation in Laplace Transforms
• $\cal{L}^{-1} \left\{ F(ks) \right\}$: Dilation in Inverse Laplace Transforms
• $\cal{L}^{-1} \left\{ F(as+b) \right\}$
• Periodic Functions
• Dirac Delta Function
• $\cal{L}\left\{ t^{n}f(t) \right\}$

Systems of First-Order Linear Differential Equations

• System of First-Order Linear Differential Equations
• Picard’s Theorem

References

• Mary L. Boas, Mathematical Methods in the Physical Sciences (3rd Edition, 2008)
• William E. Boyce, Boyce’s Elementary Differential Equations and Boundary Value Problems (11th Edition, 2017)