diffeq - Math 21a Handout on Differential Equations This...

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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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development 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 quantities---numbers really. Give these measured quantities letter names such as ‘a’, ‘b’, ‘c’, etc.
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diffeq - Math 21a Handout on Differential Equations This...

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