Consider the differential equation ( dy dx =y (2-y ) )
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Consider the differential equation dxdy=y(2−y). A slope field for this equation is shown to students (graph not reproduced here).
(a) Identify all values of y at which every solution has a horizontal tangent line. Explain why these values have that property.
(b) A solution satisfies y(0)=0.5. Determine the interval of x-values over which this solution is increasing before its graph changes from increasing to decreasing. Give a reason for your answer in terms of the sign of dxdy.
(c) For the solution in part (b), determine the interval(s) of concavity on the domain you found in part (b). Support your answer by discussing how the slopes in the field change as x increases.