Three differential equations and three verbally described slope fields are listed below.
Differential Equations
I. dxdy=x−2
II. dxdy=3y
III. dxdy=21xy
Slope Fields
P. Slopes are zero on the vertical line x = 2 and increase linearly in magnitude moving left or right from that line.
Q. All slopes along a given horizontal line are equal. Their magnitude triples when the y–coordinate triples.
R. Near the axes the segments are nearly flat, but away from both axes the segments become steeper, with the first quadrant showing positive slopes and the second quadrant negative slopes of similar magnitude.
Respond to the tasks that follow.
(a) Pair each slope field P, Q, and R with the corresponding differential equation I, II, or III. Provide a detailed justification for every match you make.
(b) Consider the field that you matched with dxdy=3y. Starting at the point (0,1), sketch an approximate particular solution curve and explain, using the slope field, why the curve has the shape you draw.
(c) For the equation matched with dxdy=x−2, determine the set of all points in the plane where the slope of a solution curve is zero. Describe how these zero–slope locations manifest themselves in the corresponding slope field.