# First-Order Linear Equations

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

## Expected Educational Results

**Objective 6–1**: I can identify if a first-order ordinary differential equation is linear.

**Objective 6–2**: I rewrite a first-order linear ordinary differential equation in standard form.

**Objective 6–3**: I can solve a first-order linear ordinary differential equation.

**Objective 6–4**: I can solve a first-order linear initial value problem.

**Objective 6–5**: I identify missing solutions to a first-order linear ordinary differential equation.

## Linear Equations

$\frac{dy}{dx}+P(x)y=Q(x)$.

#### Investigation 01

Explain why an ODE in the form $\frac{dy}{dx}+P(x)y=Q(x)$ is called first-order linear.

#### Investigation 02

Evaluate: $\frac{d}{dx}\left({e}^{\int {\textstyle \phantom{\rule{-0.167em}{0ex}}}P(x)dx}\right)$

#### Investigation 03

Evaluate: $\frac{d}{dx}(f(x){e}^{\int {\textstyle \phantom{\rule{-0.167em}{0ex}}}P(x)dx})$

## CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

**Last Modified**: Wednesday, 2 September 2020 07:22 EDT