## Summary of Basics of Differential Equations

### Essential Concepts

• A differential equation is an equation involving a function $y=f\left(x\right)$ and one or more of its derivatives. A solution is a function $y=f\left(x\right)$ that satisfies the differential equation when $f$ 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 $y=y\left(x\right)$ 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 $t=0$
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 $y=f\left(x\right)$ that satisfies a given differential equation