A slope field is drawn for an unknown differential equation on the region 0 ≤ x ≤ 10,…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A slope field is drawn for an unknown differential equation on the region 0 ≤ x ≤ 10, −1 ≤ y ≤ 6. The field has these properties:
• Horizontal segments occur on the lines y = 0 and y = 4 only.
• For 0 < y < 4 all segments tilt upward to the right, with the steepest tilt occurring near y = 2.
• For y > 4 the segments tilt downward to the right, and the magnitude of the negative slope increases with distance above y = 4.
• For y < 0 the segments tilt downward to the right, and the magnitude of the negative slope increases with distance below y = 0.
A one-parameter family of differential equations is proposed: dxdy=ky(4−y), where k is a positive constant.
(a) Explain why the sign pattern of dxdy=ky(4−y) matches all four qualitative bullet points given above.
(b) Assume the proposed family is correct. For k = 0.5, solve the differential equation explicitly for y as a function of x, given the initial condition y(0) = 1. Show each step of your solution process.
(c) Use your solution from part (b) to approximate y when x = 10, expressing your answer to three decimal places.
(d) Based on your explicit solution and on features of the slope field, discuss the long-term behavior of any solution that begins with initial value 0 < y(0) < 4. Include a statement about the limit of y(x) as x → ∞.