To sketch a slope field for the differential equation dy/dx = y + 1/x, you’ll follow these steps:
- Identify the Points: Start by identifying the nine points on the axes where you will calculate the slopes. For example, you might choose points such as (1, 1), (1, 0), (1, -1), (0, 1), (0, 0), (0, -1), (-1, 1), (-1, 0), and (-1, -1).
- Calculate the Slopes: For each chosen point (x, y), compute the slope using the differential equation. Substitute the values of x and y into the equation dy/dx = y + 1/x. This will give you the slope at each point. For example:
- At (1, 1): dy/dx = 1 + 1/1 = 2
- At (1, 0): dy/dx = 0 + 1/1 = 1
- At (1, -1): dy/dx = -1 + 1/1 = 0
- At (0, 1): dy/dx = 1 + 1/0 (undefined, vertical slope)
- At (0, 0): dy/dx = 0 + 1/0 (undefined, vertical slope)
- At (0, -1): dy/dx = -1 + 1/0 (undefined, vertical slope)
- At (-1, 1): dy/dx = 1 + 1/(-1) = 0
- At (-1, 0): dy/dx = 0 + 1/(-1) = -1
- At (-1, -1): dy/dx = -1 + 1/(-1) = -2
- Draw the Slope Field: At each of the chosen points, draw a small line segment with a slope equal to the calculated value from step 2. Use a grid to ensure your points are accurately represented on a graph.
- Visualize the Direction: The slope field gives a visual representation of the direction that the solutions to the differential equation take. These line segments provide an intuitive understanding of how the functions behave around the specified coordinates.
Conclusion: By following these steps, you create a slope field that gives insight into the behavior of the differential equation dy/dx = y + 1/x. Make sure to account for any asymptotic behavior near the vertical slopes at x = 0. This field is a valuable tool for approximating solutions without needing to solve the differential equation analytically.