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Math 21a Handout on Differential Equations
This handout introduces the subject of differential equations.
But it barely scratches the
surface of this vast, growing and extremely useful area of mathematics.
1.
Differential equations in the sciences
The branch of mathematics called differential equations
is a direct application of ideas
from calculus, and as this is a mathematics course, I should begin by telling you a little bit about
what is meant by the term ‘differential equation’.
However, I’ll digress first to begin an
argument for including mathematics in the tool kit of even the most experimentally minded
scientist.
a)
Modeling in the sciences
First, I freely admit to not being an experimentalist.
In fact, until recently, I always found
the theoretical side of science much more to my liking.
Moreover, I suffered
from a fairly
common misconception:
If I only learn enough mathematics, I can uncover nature’s secrets by pure logical deduction.
I have lately come to the realization that advances in science are ultimately driven by
knowledge dug from observations and experiments.
Although logic and mathematics can say a
great deal about the suite of possible realities, only observation and experimentation can uncover
the detailed workings of our particular universe.
With the preceding understood, where is the place for mathematics in an experimentally
driven science?
The answer to this question necessarily requires an understanding of what
modern mathematics is.
In this regard, I should say that term ‘mathematics’ covers an extremely
broad range of subjects.
Even so, a unifying definition might be:
Mathematics consists of the study and development of methods for prediction
.
Meanwhile, an experimentally driven science (such as physics or biology or chemistry) has,
roughly, the following objective:
To find useful and verifiable descriptions and explanations of phenomena in the natural world
.
To be useful, a description need be nothing more than a catalogue or index.
But, an explanation
is rarely useful without leading to verifiable
predictions
.
It is here where mathematics can be a
great help.
In practice, experimental scientists use mathematics as a tool to facilitate the
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View Full Documentdevelopment of predictive explanations for observed phenomena.
And, this is how you can
profitably view the role of mathematics.
(The use of mathematics as a tool to make predictions
of natural phenomena is called modeling and the resulting predictive explanation is often called a
mathematical model
.)
At this point, it is important to realize that a vast range of mathematics has found
applications in the sciences.
One, in particular, is differential equations
.
b)
Equations
The preceding discussion about predictions is completely abstract, and so another
digression may prove useful to bring the discussion a bit closer to the earth.
In particular,
consider what is meant by a prediction:
You measure in your lab certain quantitiesnumbers
really.
Give these measured quantities letter names such as ‘a’, ‘b’, ‘c’, etc.
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 Fall '11
 GregoryJ.Galloway
 Differential Equations, Equations, Multivariable Calculus

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