Math 21a Handout on Differential EquationsThis handout introduces the subject of differential equations. But it barely scratches thesurface of this vast, growing and extremely useful area of mathematics.1. Differential equations in the sciencesThe branch of mathematics called differential equations is a direct application of ideasfrom calculus, and as this is a mathematics course, I should begin by telling you a little bit aboutwhat is meant by the term ‘differential equation’. However, I’ll digress first to begin anargument for including mathematics in the tool kit of even the most experimentally mindedscientist.a) Modeling in the sciencesFirst, I freely admit to not being an experimentalist. In fact, until recently, I always foundthe theoretical side of science much more to my liking. Moreover, I suffered from a fairlycommon 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 byknowledge dug from observations and experiments. Although logic and mathematics can say agreat deal about the suite of possible realities, only observation and experimentation can uncoverthe detailed workings of our particular universe.With the preceding understood, where is the place for mathematics in an experimentallydriven science? The answer to this question necessarily requires an understanding of whatmodern mathematics is. In this regard, I should say that term ‘mathematics’ covers an extremelybroad 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 explanationis rarely useful without leading to verifiable predictions. It is here where mathematics can be agreat help. In practice, experimental scientists use mathematics as a tool to facilitate thedevelopment of predictive explanations for observed phenomena. And, this is how you canprofitably view the role of mathematics. (The use of mathematics as a tool to make predictionsof natural phenomena is called modeling and the resulting predictive explanation is often called amathematical model.)At this point, it is important to realize that a vast range of mathematics has foundapplications in the sciences. One, in particular, is differential equations.b) EquationsThe preceding discussion about predictions is completely abstract, and so anotherdigression 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---numbersreally. Give these measured quantities letter names such as ‘a’, ‘b’, ‘c’, etc. A prediction cantake the form of a formula which determines the value for the quantity c by measuring only thequantities a and b. Such a formula might involve simply an algebraic equation which relates aand b to c.For example, if you lived in Greece some twenty five hundred years ago, you mightdiscover that the length, c, of the hypotenuse of a right triangle can be predicted from themeasured lengths, a and b, of the other two sides. Indeed, if you were Pythagoras, you wouldwrite:c = ab22+(1.1)Or, you might determine that the area, A, of a disk can be predicted with knowledge of its radius,r, using the equationA = π r2 .(1.2)These are examples of algebraic equations in that they involve simple expressionsbetween what is known (a and b in (1.1)) and what is to be predicted (c in (1.1)). A famous andmodern algebraic equation is Einstein’s formulaE = m c2(1.3)which describes how the total energy (E) of a body at rest can be computed if you know its mass(m) and the speed of light (c ≈ 3 million meters per second). A algebraic equation withapplications to biology describes how the weight of a body (say w) would change if it had weightw0 and you hypothetically scaled its length, width and height by the same factor, say s. Thisformula asserts thatw = s3 w0 .(1.4)c) Differential equations.Differential equations can arise when studying quantities which depend on someauxiliary variable. For example, it is typical in the sciences to study time dependent phenomena.A doctor can be concerned with the amount of a certain medicinal drug in the body as a functionof time. That is, there is a function which depends on the variable t = ‘time’ and its value at timet, say f(t), is the concentration of the medicine at time t in the blood.Here is another example: An environmental scientist can be concerned with theconcentration of mercury in clams along a certain stretch of river. Here, the concentration mightdepend on the distance downstream. Thus, the concern is with a function which depends on thevariable x = ‘distance downstream’ and its value at distance x, say f(x), is the concentration ofmercury in clams which are found at distance x. By the way, this concentration might depend onboth position and time--a more complicated situation which shall also concern us.Here is a third example: A developmental biologist studying fly embryos might beconcerned with the level of a certain molecular growth factor as a function of distance from theembryo head. Here, the function in question is the level of the growth factor as a function of thevariable which measures the distance from the head of the embryo. Of course, this function canalso depend on time as well as position; and it most probably does since live embryos develop astime progresses.For a fourth example, an epidemiologist might consider the number of deaths from acertain disease as a function of age at death. Here, the variable is the age, α, at death, and thenumber of deaths of people at age α from the disease gives the function. One could denote thelatter by N(α). By the way, this example illustrates an important point: The variable in questionneed not be time nor a position, but some entirely different quantity. Indeed, the sameepidemiologist might consider the average number of heart attack victims in a particular locale,as a function of the level of cholesterol in the victim. In this case, the variable, call it c, is
View Full Document
Unlocking...