Example 2: Sketching a Slope Field

Consider the differential equation $\frac{\mathit{dy}}{\mathit{dx}}={(y–1)}^3\cos \left(\mathit{\pi x}\right)$.

Sketch a slope field for the given differential equation at the nine points indicated on the axis provided below.

Identifier: The question itself is directly instructing you to sketch a “slope field”.

Step 1: Choose any one of the given points, ( x , y ),  marked by a black dot on your given x–y axis.

This example has 9 dots that you will need to find the slope at.

1) (-1,1)

2) (-1,0)

3) (-1,-1)

4) (0,1)

5) (0,0)

6) (0,-1)

7) (1,1)

8) (1,0)

9) (1,-1)

Step 2: Plug the values from the point you selected in Step 1 into your given differential equation, ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(x,y)}$.

$\frac{\mathit{dy}}{\mathit{dx}}={(y–1)}^3\cos \left(\mathit{\pi x}\right)$

1)    ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(–1,1)}={\left(\right(1)–1)}^3\cos \left(\pi \right(–1\left)\right)=0$

2)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(–1,0)}={\left(\right(0)–1)}^3\mathrm{cos(}\pi (–1\left)\right)=1$

3)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(–1,–1)}={\left(\right(–1)–1)}^3\cos (\pi (–1))=8$

4)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(0,1)}={\left(\right(1)–1)}^3\cos \left(\pi \right(0\left)\right)=0$

5)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(0,0)}={\left(\right(0)–1)}^3\cos (\pi (0\mathrm{))}=–1$

6)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(0,–1)}={\left(\right(–1)–1)}^3\cos (\pi \mathrm{(}0\mathrm{))}=–8$

7)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(1,1)}={\left(\right(1)–1)}^3\cos \left(\pi \right(1\left)\right)=0$

8)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(1,0)}={\left(\right(0)–1)}^3\cos \left(\pi \right(1\left)\right)=1$

9)  ${\frac{\mathit{dy}}{\mathit{dx}}|}_{(1,–1)}={\left(\right(–1)–1)}^3\cos \left(\pi \right(1\left)\right)=8$

Step 3: Use the value you found for the slope in Step 2 to sketch a short line with that slope at the point you chose in Step 1 .

Final Result:

After finding the slopes at each of the 9 dots on the given axis, the slope field would look like the following: