To sketch a slope field for the differential equation dydx = y + 1/x, we first need to understand the relationship between the variables. The slope field gives us a graphical representation of the slopes of the solution curves at given points in the plane.
1. **Identify Points**: We begin by identifying the nine points where we need to calculate the slopes. These points can be any chosen points, such as (-1, 0), (-1, 1), (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), and (3, 0).
2. **Calculate Slopes**: For each of these points, we substitute the x and y values into the right-hand side of the differential equation dydx = y + 1/x:
- At (-1, 0): dydx = 0 + 1/(-1) = -1.
- At (-1, 1): dydx = 1 + 1/(-1) = 0.
- At (0, 0): Here, 1/x becomes undefined, so we can’t determine a slope.
- At (0, 1): Again, 1/x is undefined.
- At (1, 0): dydx = 0 + 1/1 = 1.
- At (1, 1): dydx = 1 + 1/1 = 2.
- At (2, 0): dydx = 0 + 1/2 = 0.5.
- At (2, 1): dydx = 1 + 1/2 = 1.5.
- At (3, 0): dydx = 0 + 1/3 = 0.33 (approx.).
3. **Draw the Slopes**: Next, at each identified point, draw a short line segment with the calculated slope. Make sure to represent the slopes accurately according to the computed values.
By following these steps, we will have a visual representation of the slopes, which provides insight into how the function changes and behaves across the specified points. This slope field is helpful for understanding the solutions of the differential equation even without solving it explicitly.