Consider a differential equation of the form dy/dt = f (y), an autonomous equation, and assume that the function f (y) is continuously differentiable.
(a) Suppose y1(t) is a solution and y1(t) has a local maximum at t = t0. Let y0 = y1(t0). Show that f (y0) = 0.
(b) Use the information of part (a) to sketch the slope field along the line y = y0 in the ty-plane.
(c) Show that the constant function y2(t) = y0 is a solution (in other words, y2(t) is an equilibrium solution).
(d) Show that y1(t) = y0 for all t.
(e) Show that if a solution of dy/dt = f (y) has a local minimum, then it is a constant function; that is, it also corresponds to an equilibrium solution.
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