80%(5)4 out of 5 people found this document helpful
This preview shows page 1 - 3 out of 57 pages.
Chapter6Differential Equationsand MathematicalModelingOne way to measure how light in the ocean di-minishes as water depth increases involvesusing a Secchi disk. This white disk is 30centimeters in diameter, and is lowered into theocean until it disappears from view. The depth of thispoint (in meters), divided into 1.7, yields the coeffi-cient k used in the equation lxl0ekx. This equationestimates the intensity lx of light at depth x using l0,the intensity of light at the surface.In an ocean experiment, if the Secchi disk disap-pears at 55 meters, at what depth will only 1% ofsurface radiation remain? Section 6.4 will help youanswer this question.320
Section 6.1Slope Fields and Euler’s Method321Chapter 6 OverviewOne of the early accomplishments of calculus was predicting the future position of aplanet from its present position and velocity. Today this is just one of a number of occa-sions on which we deduce everything we need to know about a function from one of itsknown values and its rate of change. From this kind of information, we can tell how long asample of radioactive polonium will last; whether, given current trends, a population willgrow or become extinct; and how large major league baseball salaries are likely to be inthe year 2010. In this chapter, we examine the analytic, graphical, and numerical tech-niques on which such predictions are based.Slope Fields and Euler’s MethodDifferential EquationsWe have already seen how the discovery of calculus enabled mathematicians to solveproblems that had befuddled them for centuries because the problems involved movingobjects. Leibniz and Newton were able to model these problems of motion by using equa-tions involving derivatives—what we call differential equationstoday, after the notation ofLeibniz. Much energy and creativity has been spent over the years on techniques for solv-ing such equations, which continue to arise in all areas of applied mathematics.6.1What you’ll learn about• Differential Equations• Slope Fields• Euler’s Method. . . and why Differential equations have alwaysbeen a prime motivation for thestudy of calculus and remain soto this day.EXAMPLE 1Solving a Differential EquationFind all functions ythat satisfy dy dx sec2x2x 5.SOLUTIONWe first encountered this sort of differential equation (called exactbecause it gives thederivative exactly) in Chapter 4. The solution can be any antiderivative of sec2x 2x 5,which can be any function of the form y tanx x2 5x C. That family of func-tions is the generalsolution to the differential equation.Now try Exercise 1.Notice that we cannot find a unique solution to a differential equation unless we are givenfurther information. If the general solution to a first-order differential equation is continuous,the only additional information needed is the value of the function at a single point, called aninitial condition. A differential equation with an initial condition is called an initial valueproblem. It has a unique solution, called the