MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
For the differential equation dxdy=m(4−y)(y+1), the parameter m is a positive constant. You will examine how varying m affects the qualitative features of the slope field.
(1) Part 1: (a) Determine all equilibrium solutions for any positive m and classify each as stable or unstable, citing evidence from the differential equation.
(2) Part 2: (b) On a single set of axes that extends from y=−2 to y=5, sketch two representative slope fields: one for m=0.5 (use dashed segments) and one for m=3 (use solid segments). Label each equilibrium and indicate the direction of motion of solution curves near them.
(3) Part 3: (c) Using your sketches, estimate the value of y when x=1 for the solution satisfying y(0)=−0.5 in each of the two cases. Explain the reasoning behind each estimate.
(4) Part 4: (d) Describe what happens to the appearance of the slope field as m→∞, and explain the effect on the time required for a solution starting at y=−0.5 to reach y=3.