Differential Equations

Study of equations involving derivatives and their applications.

What Are Differential Equations?First-Order Differential EquationsSecond-Order Differential Equations
Systems of Differential EquationsPartial Differential EquationsNonlinear Differential Equations and Chaos
Modeling Population GrowthEngineering Vibrations and OscillationsMedical and Biological Applications
How to Approach Differential Equation ProblemsTips for Test Day Success
What Are Differentia...First-Order Differen...Second-Order Differe...Systems of Different...Partial Differential...Nonlinear Differenti...Modeling Population ...Engineering Vibratio...Medical and Biologic...How to Approach Diff...Tips for Test Day Su...
Basic Concepts

What Are Differential Equations?

Understanding The Basics

Differential equations are equations that involve derivatives, which show how things change. Rather than just finding a number, you're often trying to find a whole function that fits a certain rule about its rate of change.

Types of Differential Equations

  • Ordinary Differential Equations (ODEs): Only involve one variable and its derivatives.
  • Partial Differential Equations (PDEs): Involve multiple variables and their partial derivatives.

Why Do We Care?

Differential equations help us model and predict real-life phenomena, from how a disease spreads to how planets move in space!

Real-World Connection

Any process that changes over time, like the cooling of a hot drink or the growth of a plant, can often be described by a differential equation.

Examples

  • The equation \( \frac{dy}{dt} = ky \) models exponential growth (like populations).

  • Newton’s law of cooling: \( \frac{dT}{dt} = -k(T - T_{env}) \) models how objects cool down.

In a Nutshell

Differential equations describe how things change and allow us to predict the future behavior of systems.

Key Terms

Derivative
A measure of how a function changes as its input changes.
Ordinary Differential Equation (ODE)
An equation involving derivatives of a function with respect to one variable.
Partial Differential Equation (PDE)
An equation involving partial derivatives with respect to more than one variable.
Next
First-Order Differential Equations