Differential Equations

Why

We want to model physical phenomena.

Definition

A differential equation is an equation relating functions and their derivatives.

Basic example

For example, let $f: \R \to \R $ be differentiable everywhere and denote the derivative of $f$ by $f': \R \to \R $. Suppose that there exists $\alpha \in \R $ so that for all $x \in \R $,

\[ f'(x) = \alpha f(x). \]

Solutions

The solution of a differential equation is a function (or functions) which satisfy the equation.¹

1. Future editions will expand. ↩︎