hamming

14 CHAPTER 2Figure 2.Ias the reduced form for the growth problem, where the saturation level is now 1. Separation of variablesplus partial fractions yields:A is, of course, determined by the initial conditions, where you put t (or x)=0. You see immediately the “S”shape of the curve; at t =–∞, z=0; at t=0, z=A/(A+1); and at t =+∞, z=1.A more flexible model for the growth is (in the reduced variables)This is again a variables separable equation, and also yields to numerical integration if you wish. We cananalytically find the steepest slope by differentiating the right hand side and equating to 0. We getHence at the placewe have the maximum slopeA direction field sketch Figure 2.I will often indicate the nature of the solution and is particularly easy to doas the slope dependsonly on y and not on x—the isoclines are horizontal lines so the solution can be slid along the x-axiswithout changing the “shape” of the solution. For a given a and b there is really only one shape, and the