Direction fields

 

For the differential equation \[ y' = x - y \] (a) Plot a direction field in the rectangle R[\( (-4 \lt x \lt 4 ) \) \( (-4 \lt y \lt 4 ) \)].
(b) Sketch some solution curves on the direction field.

A direction field shows points where \[ y' = x - y = \text{constant}\]

Vary x and y to get y' at different points.

Calculate y' at points in the rectangle R.

Here are the values of y' at points in the upper right quadrant.

  

Plot the direction field in the rectangle R.

Here is the plot

  

Identify areas of interest on the direction field.

When y' is initially one it remains one.
When y' is initially less than one, it increases until it becomes one.
When y' is initially greater than one, it decreases until it becomes one.

Here are some solution curves:

  

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

For the differential equation \[ y' = xy \] (a) Plot a direction field in the rectangle R[\( (-3 \lt x \lt 3 ) \) \( (-3 \lt y \lt 3 ) \)].
(b) Sketch some solution curves on the direction field.