applied calculus
Differential equations express how quantities change.
Definition
A differential equation is an equation involving a function and one or more of its derivatives. Such equations describe how a quantity changes in relation to itself or to other variables.
Why they matter
Differential equations appear in physics, biology, economics, and engineering. They can be used tomodel processes such as population growth, cooling, motion, and electrical circuits.
Simple example
The equation dy/dx = ky describes exponential growth or decay. If k > 0, the quantity grows; if k < 0, it decays. This is one of the simplest and most useful models in applied mathematics.
Types
There are many kinds of differential equations, including first-order equations, second-order equations, linear equations, and separable equations. Different types often require different techniques for solving them.
Key idea
Differential equations are not just symbolic puzzles. They express change in a language that can model real systems. The goal is often not only to solve them, but to interpret what the solution means in context.