tailieunhanh - Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 53

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 53. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Introducing Differential Equations 501 General and Particular Solutions to Differential Equations The families of functions we found as solutions to the differential equations in Example are not simply more general solutions but in fact are the general solutions to each of the differential equations. By this we mean that any particular solution to the differential equation can be expressed in this form. Terminology The graph of a solution is called a solution curve. The table below gathers together examples of some differential equations and their solutions. Differential Equation Some Particular Solutions The General Solution Some Solution Curves y dy n i y 7 r - dT 0 y C d ly -12 - dy 3 dx y 3x y 3x 2 y 3x C dr ky y 5ekt y -2ekt y Cekt In this section we will focus primarily on the differential equation ky. It is ubiquitous arising in fields as varied as economics biology and chemistry. 502 CHAPTER 15 Take It to the Limit The differential equation ky has the general solution dt y t Cekt where C is an arbitrary constant. EXERCISE EXAMPLE SOLUTION In order to pick out a particular solution it is only necessary to know one point on the solution This is equivalent to being given exactly one data point that is one value of the independent variable along with the corresponding value of the dependent variable. Such a piece of information is called an initial condition. Let k be an arbitrary constant. Show that for any point P in the plane there is a value of C such that the curve y Cekt passes through P. For each of the differential equations below flnd the particular solution corresponding to the initial condition given. a ddr -0-22 where 2 0 10 b dW 3W where W 2 -7 a The general solution is 2 f Ce 2f. To flnd the particular solution we use the information that 2 1 when f . 1 Ce 2 1 C The particular solution is 2 f 1 e 2f. b This differential equation is basically of the same form as the one in part a . The role of t is now played by x the .