Summary of Basics of Differential Equations

A differential equation is an equation involving a function [latex]y=f\left(x\right)[/latex] and one or more of its derivatives. A solution is a function [latex]y=f\left(x\right)[/latex] that satisfies the differential equation when [latex]f[/latex] and its derivatives are substituted into the equation.
The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.
A differential equation coupled with an initial value is called an initial-value problem. To solve an initial-value problem, first find the general solution to the differential equation, then determine the value of the constant. Initial-value problems have many applications in science and engineering.

Glossary

differential equation: an equation involving a function [latex]y=y\left(x\right)[/latex] and one or more of its derivatives

general solution (or family of solutions): the entire set of solutions to a given differential equation

initial value(s): a value or set of values that a solution of a differential equation satisfies for a fixed value of the independent variable

initial-value problem: a differential equation together with an initial value or values

order of a differential equation: the highest order of any derivative of the unknown function that appears in the equation

particular solution: member of a family of solutions to a differential equation that satisfies a particular initial condition

solution to a differential equation: a function [latex]y=f\left(x\right)[/latex] that satisfies a given differential equation