Question

Logistic Equation The logistic differential equation y′=y(1−y) appears often in problems such as population modeling.

(a) Graph the slope field of the differential equation between y= 0 and y= 1. Does the slope depend on t?

(b) Suppose f is a solution to the initial value problem with f(0) = 1/2. Using the slope field, what can we say about fast→∞? What can we say about fast→−∞?

(c) Verify that f(t) =11 +e−tis a solution to the initial value problem from part

(b). As t→ ±∞, is the end behavior of the solution similar to what you expected from the slope field above?

(d) Now, explicitly solve the initial value problem

Dy/dt=y(1−y)         y(0) =1/2

Hint: The logistic equation is separable. You may need to use partial fraction decomposition.

Answer 1