Page 29 - Schaum's Outline of Differential Equations

12 AN INTRODUCTION TO MODELING AND QUALITATIVE METHODS [CHAR 2

If P(t) is close to 100,000 (meaning that 100,000 -P(t) ~0), then the differential equation can be approxi-
mated as P'(t) ~ kP(t)(0) = 0. An approximate solution to this is P(t) = 100,000, since only a constant has a deriva-
tive equal to 0. So "in the large", P(t) "levels off" to 100,000, the carrying capacity of the population.
In this problem, we used a qualitative approach: we were able to decipher some information and express it in
a descriptive way, even though we did not possess the solution to the differential equation. This type of equation is
an example of a logistic population model and is used extensively in sociological studies. Also see Problem 7.7.

2.11. Sometimes differential equations are "coupled" (see Chapter 17 and Chapter 25); consider the following
system:

Here, let R represent the number of rabbits in a population, while F represents the number of foxes, and
t is time (months). Assume this model reflects the relationship between the rabbits and foxes. What does
this model tell us?
This system of equations (1) mirrors a "predator-prey" relationship. The RF terms in both equations can be
interpreted as an "interaction term". That is, both factors are needed to have an effect on the equations.
We see that the coefficient of R in the first equation is +2; if there was no RF term in this equation, R would
increase without bound. The -3 coefficient of RF has a negative impact on the rabbit population.
Turning our attention to the second equation, we see that F is multiplied by a - 4, indicating that the fox
population would decrease if they did not interact with rabbits. The positive coefficient for RF indicates a positive
impact on the fox population.
Predator-prey models are used extensively in many fields ranging from wildlife populations to military strategic
planning. In many of these models qualitative methods are employed.

Supplementary Problems

2.12. Using Problem 2.1. find a model which converts temperatures from degrees on the Fahrenheit scale to degrees on
the Celsius scale.

2.13. Charles' law states that, for an ideal gas at a constant pressure, — = k, where V (liters), T (degrees Kelvin) and k is
a constant (lit/°K). What does this model tell us?
2.14. Discuss Newton's second law of motion:
2.15. Suppose a room is being cooled according to the model where t (hours) and T (degrees Celsius).
If we begin the cooling process at t = 0, when will this model no longer hold? Why?
2.16. Suppose the room in Problem 2.15. was being cooled in such a way that where the variables
and conditions are as above. How long would it take for the room to cool down to its minimum temperature? Why?
2.17. Consider the model discussed in Problem 2.5. If we assume that the system is both "undamped" and "unforced",

that is F(t) = 0 and a = 0, the equation reduces to we letm = 1 and k = 4 for further simplicity, we
have Suppose we know that y(t) = sin 2t, satisfies the model. Describe the motion of displacement, y(t).

2.18. Consider the previous problem. Find (a) the velocity function; (b) the acceleration function.
2.19. Consider the differential equation Describe (a) the behavior of y at y = 1 and y = 2; (b) what
lappens to y if y < 1; (c) what happens to y if 1 < y < 2; (d) what happens to y if y > 2.