Consider the differential equation dy/dx = y(a by), where a and b are positive constants.
(a) Either by inspection or by the method suggested in Problems 37– 40, find two constant solutions of the DE.
(b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = ϕ(x) is increasing. Find intervals on which y = ϕ(x) is decreasing.
(c) Using only the differential equation, explain why y = a/2b is the y-coordinate of a point of inflection of the graph of a non constant solution y = ϕ(x).
(d) On the same coordinate axes, sketch the graphs of the two constant solutions found in part (a). These constant solutions partition the xy-plane into three regions. In each region, sketch the graph of a nonconstant solution y = ϕ(x) whose shape is suggested by the results in parts (b) and (c).
Data from problem 37-40
use the concept that y = c, -∞ < x < ∞, is a constant function if and only if y’ = 0 to determine whether the given differential equation possesses constant solutions.
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