MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Consider the differential equation dxdy=y2−4. A slope field on −3≤x≤3, −4≤y≤4 shows horizontal arrows along the lines y=2 and y=−2. Between these two lines arrows tilt downward; outside this band they tilt upward.
(a) Identify all constant (equilibrium) solutions and classify each as stable or unstable, using evidence from the slope field.
(b) For the integral curve passing through (0,1), determine on which x–intervals the curve is concave up and on which it is concave down. Justify your answer by discussing how the slopes change as you move vertically within the field.
(c) Use qualitative reasoning to show that the integral curve through (0,1) approaches y=2 as x→∞ but approaches y=−2 as x→−∞. Explain why it can never cross either horizontal line.
(d) Three points are marked in the field: P(−1,1), Q(1,1), and R(0,3). Rank the magnitude of the slopes at these points from least to greatest and justify your ranking using features of the differential equation or slope field.