Differential Equations

Study of equations involving derivatives and their applications.
What are Differential Equations?First-Order Differential EquationsInitial Value Problems
Second-Order Differential EquationsSystems of Differential EquationsPartial Differential Equations (PDEs)
Modeling Population GrowthPhysics: Motion and ForcesMedicine: Drug Dosage and Decay
Break Down the ProblemCheck Your Solution
What are Differentia...First-Order Differen...Initial Value Proble...Second-Order Differe...Systems of Different...Partial Differential...Modeling Population ...Physics: Motion and ...Medicine: Drug Dosag...Break Down the Probl...Check Your Solution
Basic Concepts

What are Differential Equations?

Introduction

Differential equations are mathematical equations that relate a function with its derivatives. In simple terms, they show how a quantity changes over time or space.

Types of Differential Equations

  • Ordinary Differential Equations (ODEs): Involve derivatives with respect to one variable.
  • Partial Differential Equations (PDEs): Involve derivatives with respect to two or more variables.

Why Are They Important?

Differential equations help us describe all sorts of changes in the world, from how planets move to how diseases spread.

Notation

  • \( \frac{dy}{dx} = f(x, y) \): Shows how \( y \) changes with \( x \).
  • \( y' = f(x, y) \): Another way to write the same thing.

Visualizing Solutions

The solution to a differential equation is a function (or set of functions) that makes the equation true.

Examples

  • A falling ball's speed is described by \( \frac{dv}{dt} = -g \) where \( g \) is gravity.

  • Population growth: \( \frac{dP}{dt} = rP \), where \( P \) is population and \( r \) is growth rate.

In a Nutshell

Differential equations describe changes and relationships using derivatives.

Key Terms

Derivative
A measure of how a function changes as its input changes.
Ordinary Differential Equation (ODE)
A differential equation involving derivatives with respect to a single variable.
Partial Differential Equation (PDE)
A differential equation involving derivatives with respect to multiple variables.
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First-Order Differential Equations
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