Summary of First-order Linear Equations | Calculus II

Module 4: Differential Equations

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