# Think Before You Calculate!

I encourage students to build their qualitative problem solving skills by recasting equations in dimensionless variables, analyzing the limiting behavior of mathematical expressions, and sketching plots showing how functions behave. “Think Before You Calculate!” is my mantra. But how, specifically, do you do this? Let me show you an example.

In Section 2.10 of *Intermediate Physics for Medicine and Biology*, Russ Hobbie and I discuss the logistic model.

Sometimes a growing population will level off at some constant value. Other times the population will grow and then crash. One model that exhibits leveling off is the

logistic model, described by the differential equation

dy/dt=b0y(1 —y/y∞) , (2.28)where

b0 andy∞ are constants….If the initial value of

yisy0, the solution of Eq. 2.28 is

y(t) = 1 / [1/y∞ + (1/y0–1/y∞) e−b0t] . (2.29)

Below is a new homework problem, analyzing the logistic equation in a way to build insight. Consider it an early Christmas present. Santa won’t give you the answer, so you need to solve the problem yourself to gain anything from this post.

Section 2.10

Problem 36 ½. Consider the logistic model.

(a) Write Eq. 2.28 in terms of dimensionless variablesYandT, whereY=y/y∞ andT=b0t.

(b) Express the solution Eq. 2.29 in terms ofY,T, andY0 =y0/y∞.

(c) Verify that your solution in part (b) obeys the differential equation you derive in part (a).

(d) Verify that your solution in part (b) is equal toY0 atT= 0.

(e) In a plot ofY(T) versusT, which of the three constants (y∞,y0, andb0) affect the qualitative shape of the solution, and which just scale theYandTaxes?

(f) Verify that your solution in part (b) approaches 1 asTgoes to infinity.

(g) Find an expression for the slope of the curveY=Y(T). What is the slope at timeT= 0? For what value ofY0 is the initial slope largest? For what values ofY0 is the slope small?

(h) The plot in Fig. 2.16 compares the solution of logistic equation with the exponentialY=Y0 e. The figure gives the impression that the exponential is a good approximation to the logistic curve at small times. Do the two curves have the same value atT= 0? Do the two curves have the same slope atT= 0?

(i) Sketch plots ofYversusTforY0 = 0.0001, 0.001, 0.01, and 0.1.

(j) Rewrite the solution from part (b),Y=Y(T), using the constantT0, whereT0 = ln[(1−Y0)/Y0]. Show that varyingY0 is equivalent to shifting the solution along theTaxis. What value ofY0 corresponds toT0 = 0?

(k) How does the logistic curve behave ifY0 > 1? Sketch a plot ofYversusTforY0 =1.5.

(l) How does the logistic curve behave ifY0 < 0? Sketch a plot ofYversusTforY0 = -0.5.

(m) PlotYversusTforY0 = 0.1 on semilog graph paper.

If you solve this new homework problem and want to compare you solution to mine, email me at roth@oakland.edu and I’ll send you my solution.

*Originally published at **http://hobbieroth.blogspot.com**.*