Up: Math 222 Syllabus
Problems on Modeling with Systems of DE's
For the following problems, set up but do not solve a system of
differential equations describing the problem. (There will be no initial
conditions given in these problems.)
- The gross national product (GNP) represents the sum of comsumption
purchases of goods and services, government purchases of goods and
services, and gross private investment (which is the increase in
inventories plus buildings constructed and equipment acquired). Assume
that the GNP is increasing at the rate of 3% per year and that the
national debt is increasing at a rate proportional to the GNP. Construct a
system of two ordinary differential equations modeling the interaction
between the GNP and the national debt.
- Suppose a concentration containing D mg of a drug (and a negligible
amount of carrier) is injected into the blood stream of a patient at time
t = 0. The drug flows from the blood into the tissue of the patient at a
rate F12 and in the reverse direction at a rate F21. The drug
flows out of the blood into the kidneys at a rate F01. Set up a
system of differential equations describing the amount of the drug in the
blood and the amount of the drug in the tissue at time t. Let Vb and
Vt denote the volumes of the blood and the tissue, repectively.
- Suppose two countries X and Y possess nuclear arms and each
increases its weapons as the other side increases its weapons. Let x(t)
denote the number of weapons possessed by country X and y(t) be the
number possessed by country Y. If b is a positive proportionality
constant measuring ``perceived threat'', assume that the rate of increase
of country X's forces is proportional to the level of Y's forces with
proprtionality constant b. Suppose that the situation for country Y is
symmetrical, with the perceived threat measured by constant m. Set up an
appropriate system of differential equations.
- Now add to the preceeding problem political and economic forces which
dampen the arms race. For instance, a percentage of the weapons force may
become obselete over time, requiring expenditures for upgrading the
systems. Let the nonnegative constant a measure this damping tendency
for country X and assume that the damping effect is proportional to the
current level of X's forces with proportionality constant a. Again,
suppose the situation for Y is symmetrical, with damping constant n.
Modify your answer to the previous problem to add in this damping effect.
- Let us assume a political system consists of two parties, say party
R and party D. Suppose pollsters have observed that about 30% of the
D's reregister as R's before each next election. Likewise, R's
migrate to party D at the rate of 25% between elections. Assume a
negligible number of independent voters in this system. Set up a
system of differential equations to describe the number of voters in each
party.
Problems on Modeling with Systems of DE's
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The translation was initiated by Joshua R Holden on 2003-12-08
Up: Math 222 Syllabus
Joshua R Holden
2003-12-08