## Summary of First-order Linear Equations

### Essential Concepts

• Any first-order linear differential equation can be written in the form $y^{\prime} +p\left(x\right)y=q\left(x\right)$.
• We can use a five-step problem-solving strategy for solving a first-order linear differential equation that may or may not include an initial value.
• Applications of first-order linear differential equations include determining motion of a rising or falling object with air resistance and finding current in an electrical circuit.

## Key Equations

• standard form

$y^{\prime} +p\left(x\right)y=q\left(x\right)$
• integrating factor

$\mu \left(x\right)={e}^{\displaystyle\int p\left(x\right)dx}$

## Glossary

integrating factor
any function $f\left(x\right)$ that is multiplied on both sides of a differential equation to make the side involving the unknown function equal to the derivative of a product of two functions
linear
description of a first-order differential equation that can be written in the form $a\left(x\right){y}^{\prime }+b\left(x\right)y=c\left(x\right)$
standard form
the form of a first-order linear differential equation obtained by writing the differential equation in the form $y^{\prime} +p\left(x\right)y=q\left(x\right)$