Summary of First-order Linear Equations

Essential Concepts

  • Any first-order linear differential equation can be written in the form [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex].
  • 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

    [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex]
  • integrating factor

    [latex]\mu \left(x\right)={e}^{\displaystyle\int p\left(x\right)dx}[/latex]


integrating factor
any function [latex]f\left(x\right)[/latex] 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
description of a first-order differential equation that can be written in the form [latex]a\left(x\right){y}^{\prime }+b\left(x\right)y=c\left(x\right)[/latex]
standard form
the form of a first-order linear differential equation obtained by writing the differential equation in the form [latex]y^{\prime} +p\left(x\right)y=q\left(x\right)[/latex]