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

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

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

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

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

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

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

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

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

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

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