Computer Algebra Recipes

152 CHAPTER 4. NONLINEAR ODE MODELS
PROBLEMS:
Problem 4-1: Falling basketball
A spherical object (diameter d meters and mass m kg) falling from rest experiences
[FC99] a drag force Fdrag = ¡Av¡ Bv2 newtons, where v is the velocity
in m/s and A =1:55 £ 10 ¡4 d, B =0:22 d2 . Derive the nonlinear ODE governing
the velocity of the falling sphere. If the sphere is a basketball of diameter
25 cm and mass 0.60 kg, analytically determine v(t). Does Maple recognize the
ODE as being separable? Plot v(t) and show that the ball will reach a terminal
velocity. Determine the terminal velocity. How long does it take the basketball
to come within 1 percent of the terminal velocity?
Problem 4-2: It's separable
Consider the ODE dy
dx = 2 x3 y ¡ y4 x4 ; y(1) = 5:
¡ 2 xy3 Does this ODE appear to be separable? Show that assuming y(x) =xz(x)leads
to a separable equation for z(x). Making use of this transformation, determine
y(x) and plot the solution, starting at x = 1, over the range for which it remains
real. What is the x value at the upper end of this real range?
4.1.2 The Struggle for Existence
The mathematics of uncontrolled growth are frightening. A single
cell of the bacterium E. coli would, under ideal circumstances, divide
[in two] every twenty minutes .... it can be shown that in a single
day, one cell of E. Coli could produce a super-colony equal in size
and weight to the entire planet Earth.
Michael Crichton, The Andromeda Strain (1969)
A classic experiment in microbiology is to grow yeast, or other microorganisms,
in a nutrient broth inside a test tube or °ask at a suitable ¯xed temperature.
As an assignment associated with her microbiology course in the premed
program at MIT, Heather has been asked to create a Maple worksheet that illustrates
the solution of simple model equations describing the growth of yeast
in a test tube. She is to search the literature and ¯nd realistic numbers for
the parameter values and use these to create suitable plots of the solutions.
Consulting her older sister Jennifer who, recall, is a mathematics faculty member
at MIT, Heather is guided to the text Mathematical Models in Biology by
Leah Edelstein-Keshet [EK88]. This interesting and easy-to-read book describes
some yeast-growing experiments and associated model equations.
After loading the library plots package,
> restart: with(plots):
Heather considers the simplest model of yeast growth, which would apply if
there were an unlimited supply of nutrient. She lets N(t) be the yeast popula-