# 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

#### Investigation 04

Determine if the following ODEs are first-order linear ODEs. Explain.

1. $\frac{dy}{dx}+\frac{1}{5}y=5-x$$\displaystyle \frac{dy}{dx}+\frac{1}{5}y=5-x$, $x>0$$x>0$

2. $\frac{dy}{dx}+\frac{1}{x}y=2$$\displaystyle \frac{dy}{dx}+\frac{1}{x}y=2$, $x>0$$x>0$

3. $\frac{dy}{dx}+3{x}^{2}y=6{x}^{2}$$\displaystyle \frac{dy}{dx}+3x^2y=6x^2$

4. ${y}^{\prime }-y\mathrm{tan}\left(x\right)=1$$\displaystyle y^{\prime}-y\tan{(x)}=1$, $-\frac{\pi }{2}$\displaystyle -\frac{\pi}{2}

5. $\frac{dy}{dx}+\mathrm{cot}\left(x\right)y=x\mathrm{csc}\left(x\right)$$\displaystyle \frac{dy}{dx}+\cot{(x)}y=x\csc{(x)}$

#### Investigation 05

For each of the following ODEs, find the general solution [do not forget $+C$$+C$], if possible:

1. $\frac{dy}{dx}+\frac{1}{5}y=5-x$$\displaystyle \frac{dy}{dx}+\frac{1}{5}y=5-x$, $x>0$$x>0$

2. $\frac{dy}{dx}+\frac{1}{x}y=2$$\displaystyle \frac{dy}{dx}+\frac{1}{x}y=2$, $x>0$$x>0$

3. $\frac{dy}{dx}+3{x}^{2}y=6{x}^{2}$$\displaystyle \frac{dy}{dx}+3x^2y=6x^2$

4. ${y}^{\prime }-y\mathrm{tan}\left(x\right)=1$$\displaystyle y^{\prime}-y\tan{(x)}=1$, $-\frac{\pi }{2}$\displaystyle -\frac{\pi}{2}

5. $\frac{dy}{dx}+\mathrm{cot}\left(x\right)y=x\mathrm{csc}\left(x\right)$$\displaystyle \frac{dy}{dx}+\cot{(x)}y=x\csc{(x)}$

#### Investigation 06

For each of the following IVPs, find the particular solution, if possible:

1. ${e}^{x}\frac{dy}{dx}=20+3{e}^{x}y$$\displaystyle e^x\frac{dy}{dx}=20+3e^xy$, $y\left(1\right)=1$$y(1)=1$

2. $x{y}^{\prime }-2y=5{x}^{2}$$\displaystyle xy^{\prime}-2y=5x^2$, $y\left(1\right)=2$$y(1)=2$