# First-Order Linear Equations

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

• Section 2.6 – Linear Equations

## 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

### Definition: Standard Form of a Linear First-order DE

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

#### Investigation 01

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

#### Investigation 02

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

#### Investigation 03

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