Differential Equations

Differential Equations Curriculum

This curriculum provides a structured overview of the study of differential equations.

First-Order Ordinary Differential Equations

Introduction to Differential Equations:

• Definitions and terminology (order, linearity, etc.)
• Initial value problems
• Direction fields
• Existence and uniqueness of solutions
• Applications of differential equations

Methods for Solving First-Order ODEs:

• Separable equations
• Linear equations (integrating factors)
• Exact equations
• Homogeneous equations
• Bernoulli equations

Applications of First-Order ODEs:

• Growth and decay models
• Newton's Law of Cooling
• Mixing problems
• Population dynamics
• Circuit analysis

Second-Order Linear Ordinary Differential Equations

Homogeneous Equations:

• General solution
• Linear independence and the Wronskian
• Constant coefficients (real and complex roots)
• Repeated roots
• Reduction of order

Nonhomogeneous Equations:

• Method of undetermined coefficients
• Variation of parameters
• Cauchy-Euler equations
• Forced oscillations and resonance
• Applications of nonhomogeneous equations

Series Solutions of Linear Equations:

• Power series review
• Series solutions near an ordinary point
• Regular singular points
• Method of Frobenius
• Special functions (Bessel functions, Legendre polynomials)

Systems of Ordinary Differential Equations

First-Order Systems:

• Matrix methods
• Homogeneous linear systems
• Nonhomogeneous linear systems
• Phase plane analysis
• Applications of systems of ODEs

Applications of Systems of ODEs:

• Coupled oscillators
• Compartmental models
• Predator-prey models
• Electrical networks
• Multi-dimensional dynamics