A slope field for the differential equation dy dx =y-2x is shown on a square grid of…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field for the differential equation
dxdy=y−2x
is shown on a square grid of points with integer coordinates –2 ≤ x ≤ 2 and –3 ≤ y ≤ 3. At each lattice point a short line segment has been drawn whose slope equals y−2x. (For example, at the point (1,3) the segment is nearly horizontal because 3−2(1)=1, while at (2,3) the segment tilts downward because 3−2(2)=−1.)
Without solving the differential equation, respond to the prompts below. Base every conclusion explicitly on information that can be read from the slope field.
(a) Determine whether the solution curve that passes through the point (1,3) is increasing or decreasing at that point. Justify your conclusion by referring to the local segment in the slope field.
(b) Explain why any solution curve that intersects the line y=2x must have a horizontal tangent at the point of intersection.
(c) Describe, in interval notation, all x–values on the grid for which every solution curve is guaranteed to be decreasing. Provide reasoning that relies solely on the slopes shown in the field.