Final answer:
A slope field for the differential equation (dy)/(dx)=(xy)/(2) is created by calculating the slope at selected grid points and drawing line segments at those points reflecting the slope.
Step-by-step explanation:
To construct a slope field for the differential equation (dy)/(dx)=(xy)/(2), we need to understand that the slope is the difference in y-value (the rise) divided by the difference in x-value (the run) of two points on a straight line. We follow a step-by-step procedure to create the slope field:
- Select a grid of points within the region of interest on the plane.
- At each point (x, y), calculate the slope (xy)/(2) which is the value of (dy/dx) at that point.
- Draw a small line segment at each point with the calculated slope. This represents the solution curve passing through that point.
The slope field helps visualize the behavior of solutions to the differential equation across different points in the plane.
The complete question is: Construct a slope field for the differential equation: (dy)/(dx)=(xy)/(2) is: