MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Consider the differential equation dxdy=2−y. Grid axes with tick marks for −1≤x≤3 and −1≤y≤4 are provided in the answer booklet.
(1) Explain why every solution curve that begins at a point with y>2 is decreasing for all real values of x.
(2) Describe the long–run behavior of any solution curve that begins at a point with y<2. In particular, state whether such a curve levels off, and if so, give its limiting value and justify your conclusion without solving the differential equation.
(3) On the provided axes, sketch at least 20 line segments to create a representative slope field for the given differential equation over the indicated domain, and then draw the solution curve that passes through the point (0,0). Clearly label the curve you draw.