Let be a fixed constant and consider the parameter–dependent family of differential…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Let a>0 be a fixed constant and consider the parameter–dependent family of differential equations
dxdy=y(y−a)(b−y),
where the parameter b>a controls the upper equilibrium level.
(a) In terms of a and b, determine all equilibrium solutions and classify the stability of each.
(b) For fixed a, explain how varying b affects the vertical ordering and relative stability of the equilibria. Support your answer by discussing the sign of dxdy in the intervals determined by the equilibria.
(c) Sketch two slope fields on the same set of axes—one for b=23a and one for b=4a. Use different styles (for example, different line weights or colors) to distinguish the two fields and provide a key.
(d) A solution curve starts at the initial condition (0,43a). Determine, with justification based on your analysis in parts (a)–(b), whether increasing b shortens, lengthens, or leaves unchanged the x–interval required for the solution to reach y=89a.
(e) For b=4a, write a brief explanation of how the slope field illustrates the long-term behavior (x→∞) of all solutions that start with 0<y(0)<b.