Consider the differential equation
dxdy=x2−y.
Four slope-field descriptions, labelled A–D, are given.
A. Horizontal segments occur exactly at points satisfying y=x2. Segments are steeper and tilt upward to the right when y<x2; they tilt upward to the left when y>x2.
B. Horizontal segments occur exactly on the line y=x. Segments tilt upward to the right when y<x; they tilt upward to the left when y>x.
C. Horizontal segments occur along the x-axis. Above the x-axis segments tilt upward to the right; below the x-axis they tilt upward to the left.
D. Slopes depend only on y; segments along any horizontal line share the same tilt, while segments in a given vertical column have different tilts.
(a) Indicate which description (A, B, C, or D) matches the given differential equation.
(b) Provide two separate mathematical reasons supporting your choice in part (a). Each reason must relate a specific feature of the differential equation to an observed feature in the matching description.
(c) Let the solution curve through the point (1, 1) be denoted by y = s(x). Using the differential equation, determine whether s(x) is increasing or decreasing at x = 1 and decide whether it is concave upward or concave downward there. Confirm that these conclusions are consistent with the slope-field description you selected.