### Essential Concepts

- 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 velocity
- the velocity at time [latex]t=0[/latex]

- 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