How do you draw slope fields?

Question

Aurora Alvarado · Accepted Answer

Example:

How do you draw the slope field for #dy/dx = x - y#?

The slope field is a cartesian grid where you draw lines in various directions to represent the slopes of the tangents to the solution. Therefore by drawing a curve through consecutive slope lines, you can find a solution to the differential equation.

Take the example of #dy/dx# at #(3, 4)#. Here we see that

#dy/dx = 3 - 4 =-1#

So you would draw a line of slope #-1# at #(3,4)#. Repeat this for maybe 4 by 4 points to get the following slope field

Hopefully this helps!

Christopher Agresta · Answer

undefined To draw a slope field:

1. Choose a grid: Draw a grid on a coordinate plane. The grid should cover the region of interest and have equally spaced horizontal and vertical lines.

2. Determine the differential equation: Given a first-order ordinary differential equation (ODE) of the form dy/dx = f(x, y), identify the function f(x, y).

3. Calculate slopes: For each point (x, y) in the grid, calculate the slope at that point using the function f(x, y). This involves evaluating f(x, y) at each point to find the slope.

4. Draw arrows: At each point on the grid, draw a short line segment or arrow with the calculated slope. The direction of the arrow indicates the direction of the slope, and the length of the arrow represents the magnitude of the slope.

5. Refine the slope field: Adjust the density of arrows as needed to accurately represent the behavior of the solution curves. You may need to add more arrows in regions with rapidly changing slopes or fewer arrows in regions with relatively constant slopes.

6. Optional: Add solution curves: If you have specific initial conditions for the ODE, you can sketch solution curves on the slope field by following the direction indicated by the arrows.

7. Label axes and add titles: Finally, label the x and y axes and add any necessary titles or labels to the slope field to provide context for the graph.