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7.3 Sketching Slope Fields

1 min readfebruary 15, 2024

Jacob Jeffries

Jacob Jeffries

Jacob Jeffries

Jacob Jeffries

Slope fields allow us to visualize a solution to a differential equation without actually solving the differential equation. Let’s construct a slope field to solidify this idea. 🧠

Slope fields essentially draw the slopes of line segments that go through certain points.

Example 1

Let’s consider the following differential equation:

dydx=x+y\frac{dy}{dx} = x+y

The slope (m) at point (x,y), in this case, is just x + y, which we can put into a table for various coordinates:

y=0y=0y=1y=1y=2y=2y=3y=3
x=0x=0m=0m=0m=1m=1m=2m=2m=3m=3
x=1x=1m=1m=1m=2m=2m=3m=3m=4m=4
x=2x=2m=2m=2m=3m=3m=4m=4m=5m=5
x=3x=3m=3m=3m=4m=4m=5m=5m=6m=6

We can use this data to draw an approximate solution to the differential equation by drawing short line segments through each point that have the corresponding slope:

https://storage.googleapis.com/static.prod.fiveable.me/images/image87.png-1692660989791-26019

Source: Jacob Jeffries

Example 2

Let’s consider another differential equation:

dydx=xy\frac{dy}{dx} = \frac{x}{y}

The slope (m) at point (x,y), in this case, is just xy\frac{x}{y}, which we can put into a table for various coordinates:

y=0y=0y=1y=1y=2y=2y=3y=3
x=0x=0undefinedm=0m=0m=0m=0m=0m=0
x=1x=1undefinedm=1m=1m=12m=\frac{1}{2}m=13m=\frac{1}{3}
x=2x=2undefinedm=2m=2m=1m=1m=23m=\frac{2}{3}
x=3x=3undefinedm=3m=3m=32m=\frac{3}{2}m=1m=1

We can use this data to draw an approximate solution to the differential equation by drawing short line segments through each point that have the corresponding slope.

The graph below includes more points than in the table to provide you with a better illustration of what a slope field looks like:

Screen Shot 2023-01-01 at 2.31.58 PM.png

Source: Jacob Jeffries

To be continued in 7.4 Reasoning Using Slope Fields.

7.3 Sketching Slope Fields

1 min readfebruary 15, 2024

Jacob Jeffries

Jacob Jeffries

Jacob Jeffries

Jacob Jeffries

Slope fields allow us to visualize a solution to a differential equation without actually solving the differential equation. Let’s construct a slope field to solidify this idea. 🧠

Slope fields essentially draw the slopes of line segments that go through certain points.

Example 1

Let’s consider the following differential equation:

dydx=x+y\frac{dy}{dx} = x+y

The slope (m) at point (x,y), in this case, is just x + y, which we can put into a table for various coordinates:

y=0y=0y=1y=1y=2y=2y=3y=3
x=0x=0m=0m=0m=1m=1m=2m=2m=3m=3
x=1x=1m=1m=1m=2m=2m=3m=3m=4m=4
x=2x=2m=2m=2m=3m=3m=4m=4m=5m=5
x=3x=3m=3m=3m=4m=4m=5m=5m=6m=6

We can use this data to draw an approximate solution to the differential equation by drawing short line segments through each point that have the corresponding slope:

https://storage.googleapis.com/static.prod.fiveable.me/images/image87.png-1692660989791-26019

Source: Jacob Jeffries

Example 2

Let’s consider another differential equation:

dydx=xy\frac{dy}{dx} = \frac{x}{y}

The slope (m) at point (x,y), in this case, is just xy\frac{x}{y}, which we can put into a table for various coordinates:

y=0y=0y=1y=1y=2y=2y=3y=3
x=0x=0undefinedm=0m=0m=0m=0m=0m=0
x=1x=1undefinedm=1m=1m=12m=\frac{1}{2}m=13m=\frac{1}{3}
x=2x=2undefinedm=2m=2m=1m=1m=23m=\frac{2}{3}
x=3x=3undefinedm=3m=3m=32m=\frac{3}{2}m=1m=1

We can use this data to draw an approximate solution to the differential equation by drawing short line segments through each point that have the corresponding slope.

The graph below includes more points than in the table to provide you with a better illustration of what a slope field looks like:

Screen Shot 2023-01-01 at 2.31.58 PM.png

Source: Jacob Jeffries

To be continued in 7.4 Reasoning Using Slope Fields.



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© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.