The Gompertz equation dy/dt = y(a-b ln y) where a and b are parameters, is used in actuarial studies, and to model growth of objects as diverse as tumors and organizations.
(a) Show that the solution to the Gompertz equation is y(t) = ea/bece-bt
(b) Solve the IVP for this equation with y(0) = y0.
(c) Describe the limiting value for y(t) as t → ∞.
Assume that a > 0, and consider the cases b > 0 and b < 0.