Page 27 - Schaum's Outline of Differential Equations

10 AN INTRODUCTION TO MODELING AND QUALITATIVE METHODS [CHAR 2

QUALITATIVE METHODS

To build a model can be a long and arduous process; it may take many years of research. Once they
are formulated, models may be virtually impossible to solve analytically. Then the researcher has two
options:

• Simplify, or "tweak", the model so that it can be dealt with in a more manageable way. This is a valid
approach, provided the simplification does not overly compromise the "real-world" connection, and
therefore, its usefulness.
• Retain the model as is and use other techniques, such as numerical or graphical methods (see
Chapter 18, Chapter 19, and Chapter 20). This represents a qualitative approach. While we do not
possess an exact, analytical solution, we do obtain some information which can shed some light on the
model and its application. Technological tools can be extremely helpful with this approach (see
Appendix B).

Solved Problems

Problems 2.1 through 2.11 deal with various models, many of which represent real-world situations. Assume
the models are valid, even in the cases where some of the variables are discrete.

2.1. Discuss the model: 7> = 32 + 1.8 T c.
This model converts temperatures from degrees on the Celsius scale to degrees on the Fahrenheit scale.

2.2. Discuss the model: PV = nRT.
This models ideal gases and is known as the Perfect Gas Law. Here, P is the pressure (in atmospheres), V is the
volume (liters), n is the number of moles, R is the universal gas constant (R = 8.3145 J/mol K), and Tis the temperature
(degrees Kelvin).

2.3. What does Boyle's law tell us?
Boyle's law states that, for an ideal gas at a constant temperature, PV= k, where P (atmospheres), V (liters)
and k is a constant (atmosphere-liters).
Another way of stating this, is that the pressure and volume are inversely proportional.