The one-parameter family of differential equations dy dx =y ,(y-k) ,(2-y)
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
The one-parameter family of differential equations
dxdy=y(y−k)(2−y),k∈R,
arises in a model that depends on the real parameter k.
(a) Determine all equilibrium solutions in terms of k.
(b) For each of the following parameter regimes, classify every equilibrium solution as stable, unstable, or semi-stable. Provide a brief justification in each case.
(i) k<0(ii) 0<k<2(iii) k>2
(c) Explain how both the number of equilibrium solutions and their stability change as k passes through the critical values k=0 and k=2. Your answer should reference the behavior of dxdy and include a clear verbal description of the bifurcation that occurs.