1.3. Differential Equations as Models of Reality 7 Still, there are two reasons that readers of this book should be expected to have some experience with numerical methods. One is so that they can suﬃciently appreciate the wonderful fact that we can write exact solutions in soliton theory. The other is that the history of solitons includes some important discoveries made using these numerical methods before it was understood that exact solutions could be written down. 1.3 Differential Equations as Models of Reality and Unreality Many situations in the human and natural worlds can be modeled with differential equations. For instance, the growth of value of an investment at a fixed interest rate, the size of a population with un- limited resources, and the amount of radioactive isotope can all be described by the simple equation (1.1). More complicated equations are used to model the behavior of the immune system, the motion of objects being pulled towards each other by gravity, and the flow of air around an airplane’s wing. One can think of differential equations in those situations as rep- resenting a “law” that the objects described are believed to obey, whether it is the inverse square law of gravitational attraction or that the number of new individuals in the population is proportional to the size of the entire population. Then, checking whether a given func- tion is a solution to the equation is like checking whether it describes something that really could happen. For instance, there is a linear partial differential equation which describes how heat will move through a piece of metal. One can find a solution to that equation which shows the heat initially concentrated at the corners and slowly spreading out until the object is nearly uniform in temperature. There are also solutions in which the object starts out with a high temperature in the middle and low temperature elsewhere, and again the heat eventually evens out. That these are solutions indicates that these are ways that heat really would spread in an object. On the other hand, there are no solutions to this equa- tion showing the heat concentrating at one point in the object while the rest of the object remains cold, which is an indication that this is not possible (because it violates the law). It is also possible to be interested in differential equations as purely abstract objects, without worrying about whether they ac-

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.