Math: Slope Fields and Direction Fields
Visualizing differential equations with slopes
Math: Slope Fields and Direction Fields
Visualizing differential equations with slopes
Math - Grade 9-12
- 1
For the differential equation dy/dx = x + y, find the slope of the solution curve at the point (2, -1).
Substitute the x-coordinate and y-coordinate into the differential equation.
The slope is 1 because dy/dx = x + y = 2 + (-1) = 1 at the point (2, -1). - 2
For the differential equation dy/dx = y - 2, describe the slope field along the horizontal line y = 2.
Along the line y = 2, the slope is 0 because dy/dx = 2 - 2 = 0. The slope field has horizontal line segments everywhere on y = 2. - 3
For the differential equation dy/dx = x, explain why all points with the same x-coordinate have the same slope in the slope field.
Look at which variable appears on the right side of the equation.
All points with the same x-coordinate have the same slope because the differential equation depends only on x, not on y. For example, every point with x = 3 has slope 3. - 4
A slope field is made for dy/dx = -y. Describe what the small line segments look like above the x-axis, on the x-axis, and below the x-axis.
Test one positive y-value, y = 0, and one negative y-value.
Above the x-axis, y is positive, so the slopes are negative. On the x-axis, y = 0, so the slopes are 0 and the segments are horizontal. Below the x-axis, y is negative, so the slopes are positive. - 5
For the differential equation dy/dx = x - y, calculate the slopes at the points (0, 0), (1, 0), (1, 1), and (0, 2).
At (0, 0), the slope is 0 - 0 = 0. At (1, 0), the slope is 1 - 0 = 1. At (1, 1), the slope is 1 - 1 = 0. At (0, 2), the slope is 0 - 2 = -2. - 6
The differential equation dy/dx = 3 has a constant slope field. Describe the slope field and the general shape of its solution curves.
A constant derivative means the graph changes at a constant rate.
The slope field has the same positive slope of 3 at every point. The solution curves are straight lines with slope 3, so they have the form y = 3x + C. - 7
For dy/dx = y(4 - y), identify the equilibrium solutions and explain how they appear in the slope field.
Equilibrium solutions occur where dy/dx = 0.
The equilibrium solutions are y = 0 and y = 4 because y(4 - y) = 0 at those values. In the slope field, these appear as horizontal line segments along the lines y = 0 and y = 4. - 8
A solution curve to a differential equation passes through (0, 1). The slope field near that point has positive slopes that get steeper as x increases. Describe the likely behavior of the solution curve just to the right of x = 0.
Just to the right of x = 0, the solution curve should rise. Since the slopes get steeper as x increases, the curve should increase at a faster rate as it moves to the right. - 9
For the differential equation dy/dx = x + 1, sketching is not required. State where the slope field has horizontal segments and explain your reasoning.
Horizontal segments have slope 0.
The slope field has horizontal segments where dy/dx = 0. Since x + 1 = 0 when x = -1, the horizontal segments occur along the vertical line x = -1. - 10
The slope field for a differential equation shows horizontal segments along y = 1. Above y = 1, the slopes are positive. Below y = 1, the slopes are negative. Which differential equation best matches this description: dy/dx = y - 1, dy/dx = 1 - y, or dy/dx = x - 1? Explain your choice.
The best match is dy/dx = y - 1. This equation gives slope 0 when y = 1, positive slopes when y is greater than 1, and negative slopes when y is less than 1.