J. David Logan - A First Course in Differential Equations

Springer International Publishers, Switzerland 2015

J. David Logan 1

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Readers are familiar with solving algebraic equations. For example, the solutions to the quadratic equation

are easily found to be and , which are numbers . An ordinary differential equation, or just differential equation , is another type of equation where the unknown is not a number, but a function . We call the unknown function and think of it as a function of time . Simply put, a differential equation is an equation that relates the unknown function to some of its derivatives, which, of course, are not known either.

Why are differential equations so important that they deserve an entire course, or even lifetime, of study? Well, differential equations arise naturally as mathematical models in areas of science, engineering, economics, and many other subjects. Physical systems, biological systems, and economic systems, and so on are marked by change, or dynamics . Differential equations describe those dynamical changes. The unknown function , often called the state function , could be the distance along a line, the current in an electrical circuit, the concentration of a chemical undergoing reaction, the population of an animal species in an ecosystem, or the demand for a commodity in a microeconomy. Differential equations represent laws that govern change, and the unknown state for which we solve, describes how the changes occur. The bottom line is that most laws of nature, and other systems, relate the rate, or derivative, at which some quantity changes to the quantity itselfthus a differential equation.

In this text we set up and solve differential equations. Often the solution to an equation is given by a general formula, and some may want to memorize the formulas. Students should realize that this strategy is inappropriate; it leads to little understanding and memorized formulas fade quickly from memory. What is important in this subject is conceptual; that is, the understanding of its origin and the process of solving the equation and interpreting the results.

Historically, differential equations dates to the mid-seventeenth century when the calculus was developed by Isaac Newton (c. 1665) in the context of studying the laws of mechanics and the motion of planets (published in his Principia , 1687). In fact, some might say that calculus was invented to describe how objects move. Other famous mathematicians and scientists of that era, for example, Leibniz, the Bernoullis, Euler, Lagrange, and Laplace, contributed important results into the 1700s, and Cauchy developed some of the first theoretical concepts in the 1800s. Differential equations is now the principal tool for applications in all areas of mechanics, thermodynamics, electromagnetic theory, quantum theory, and so on. It continues today with the study of dynamical systems and nonlinear phenomena in biology, chemistry, economics, and almost every area where the dynamics of systems is important. By studying differential equations, students are rewarded with a knowledge of one of the monuments of mathematics and science, and they see the great connection between nature and mathematics in ways that they may never have imagined.