Final answer:
The differential equation dy/dx = x²/y requires sketching a slope field by calculating the slope at various points and then drawing small line segments to represent the slopes. Connecting these dashes smoothly can depict the solution curves, which will tend to follow parabolic paths.
Step-by-step explanation:
The question involves the differential equation dy/dx = x2/y. To sketch the slope field for this differential equation, you would first calculate the slope (dy/dx) at various points (x,y) in the plane. This will give you a direction or slope of the tangent to the curve at each point. Then, on graph paper, you would draw small line segments representing the slope at these points.
To show the trends and reasonably connect the dashes to illustrate potential solution curves, you would start at a point and then follow the direction of the slope field, connecting the dashes smoothly to approximate the curves that solutions to the differential equation would follow. At points where y=0, the differential equation is undefined, and thus the slope field will not have any dashes and the solution curves cannot cross this line.
The solution curves will tend to be parabolic, since when x is positive, dy/dx is positive and when x is negative, dy/dx is negative. The curvature of these parabolas will depend on the value of y; the larger the value of y, the shallower the curve, and vice versa.