Page 28 - Schaum's Outline of Differential Equations

CHAP. 21 AN INTRODUCTION TO MODELING AND QUALITATIVE METHODS 11

2.4. Discuss the model:

This formula is used in electricity; I represents the current (amperes), q represents the charge (coulombs), t is
the time (seconds). Problems involving this model will be presented in both Chapter 7 and Chapter 14.

2.5. Discuss the model:

This is a classic model: a forced, mass-spring system. Here, y is a displacement (m), t is time (sec), m is the mass
2
(kg), a is a friction or damping constant (kg/sec), k is a spring constant (kg/sec ) and F(t) is a forcing function (N).
Variations of this model can be used in problems ranging from shock absorbers on an automobile to answering
questions about the human spinal column.
The differential equation uses a number of classical concepts, including Newton's second law and Hooke's law.
We will revisit this equation in Chapter 14.

2.6. Assume M(f) represents the mass of an element in kgs. Suppose research has shown that the instanta-
neous rate of decay of this element (kg/yr) is proportional to the amount present: M(t) x M(t). Set up a
model for this relationship.
The proportionality relationship M'(t) °= M(t) can be converted into an equation by introducing a
proportionality constant, k (1/yr). So our model becomes M'(t) = kM(t). We note that k < 0, because M(t) is decreasing
in size.
This equation will be classified as a "separable equation" (see Chapter 3). The solution to this differential
equation, which is qualitatively described as "exponential decay", will be explored in Chapter 4.

2.7. Consider the previous problem. Assume research revealed that the rate of decay is proportional to the
square root of the amount present. Model this situation.
We note here that the units of k are The solution of this type
of differential equation will be explored in Chapter 4.

2.8. Model a population P(t), if its rate of growth is proportional to the amount present at time t.
This is the sister problem to Problem 2.6; that is, we have an "exponential growth" model, P'(t) = kP(t),
where k > 0.

2.9. Assume the population described in Problem 2.8 has an initial composition of 1000. That is,
F(0) = 1000. You are also told that the solution of the differential equation P'(t) = kP(t) is given by
kt
P(f) = I000e , where t is in years. Discuss this model.
Since k > 0, we know that P(t) will increase exponentially as t —> °°. We are forced to conclude that this is
(most probably) not a reasonable model, due to the fact that our growth is unlimited.
We do add, however, that this model might be helpful over a short period of time. "How helpful?" and "How
short a period?" are questions which must be looked at qualitatively, and depend on the constraints and requirements
of the particular posed problem.

2.10. Consider the assumptions in the two previous problems. Further, suppose the rate of growth of P(t)
is proportional to the product of the amount present and some "maximum population" term, 100,000 -
P(f), where the 100,000 represents the carrying capacity. That is, P(f) -» 100,000, as t -» °°. Introduction
of a proportionality constant k, leads to the differential equation, P'(t) = kP(t)(lOO,000 - P(t)). Discuss
this model.
If P(t) is much less than 100,000, the differential equation can be approximated as P'(t)~kP(t)
(100,000) = KP(t), where K= £(100,000). This would closely approximate exponential growth. So, for "small" P(t),
there would be little difference between this model and the previous model discussed in Problems 2.8 and 2.9.