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

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

Investigation 01

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

Investigation 02

Evaluate: ${\displaystyle \frac{d}{dx}\left(e^{\int P(x)dx}\right)}$$\displaystyle \frac{d}{dx} \left( e^{\int{\!P(x)dx}}\right)$

Investigation 03

Evaluate: ${\displaystyle \frac{d}{dx}(f(x)e^{\int P(x)dx})}$$\displaystyle \frac{d}{dx} \left(f(x) e^{\int{\!P(x)dx}}\right)$

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