“main”2007/2/16page 104iiiiiiii104 CHAPTER 1 First-Order Differential Equations16. The following initial-value problem arises in the anal-ysis of a cable suspended between two fixed pointsy=1a 1 + (y)2,y(0) = a, y(0) = 0,where a is a nonzero constant. Solve this initial-valueproblem for y(x). The corresponding solution curveis called a catenary.17. Consider the general second-order linear differentialequation with dependent variable missing:y+ p(x)y= q(x).Replace this differential equation with an equivalentpair of first-order equations and express the solutionin terms of integrals.18. Consider the general third-order differential equationof the formy= F(x,y). (1.11.27)(a) Show that Equation (1.11.27) can be replaced bythe equivalent first-order systemdu1dx= u2,du2dx= u3,du3dx= F(x,u3),where the variables u1,u2,u3are defined byu1= y, u2= y,u3= y.(b) Solve y= x−1(y− 1).19. A simple pendulum consists of a particle of mass msupported by a piece of string of length L. Assumingthat the pendulum is displaced through an angle θ0radians from the vertical and then released from rest,the resulting motion is described by the initial-valueproblemd2θdt2+gLsin θ = 0,θ(0) = θ0,dθdt(0) = 0.(1.11.28)(a) For small oscillations, θ<<1, we can use theapproximation sin θ ≈ θ in Equation (1.11.28) toobtain the linear equationd2θdt2+gLθ = 0,θ(0) = θ0,dθdt(0) = 0.Solve this initial-value problem for θ as a functionof t. Is the predicted motion reasonable?(b) Obtain the following first integral of (1.11.28):dθdt=±2gL(cos θ − cos θ0). (1.11.29)(c) Show from Equation (1.11.29) that the time T(equal to one-fourth of the period of motion) re-quired for θ to go from 0 to θ0is given by theelliptic integral of the first kindT =L2gθ001√cos θ − cos θ0dθ. (1.11.30)(d) Show that (1.11.30) can be written asT =Lgπ/2011 − k2sin2udu,where k = sin(θ0/2).[Hint: First express cos θand cos θ0in terms of sin2(θ/2) and sin2(θ0/2).]1.12Chapter ReviewBasic Theory of Differential EquationsThis chapter has provided an introduction to the theory of differential equations. Adifferential equation involves one or more derivatives of an unknown function, and thehighest-order derivative is the order of the differential equation.For an nth-order differential equation, the general solution contains n arbitrary con-stants, and all solutions can be obtained by assigning appropriate values to the constants.This chapter is concerned mainly with first-order differential equations, which maybe written in the form“main”2007/2/16page 105iiiiiiii1.12 Chapter Review 105dydx= f(x, y), (1.12.1)for some given function f . If we impose an initial condition specifying the value of asolution y(x) to the differential equation (1.12.1) at a particular point x0, say y0= y(x0),then we have an initial-value problem:dydx= f(x, y), y(x0) = y0. (1.12.2)To solve an initial-value problem of the form (1.12.2), the first step is to determine thegeneral solution to the differential equation (1.12.1), and then use the initial condition todetermine the specific value of the arbitrary constant appearing in the general solution.Solution Techniques for First-Order Differential EquationsOne of our main goals in this chapter is to find solutions to first-order differential equa-tions of the form (1.12.1). There are various ways in which we can seek these solutions:1. Geometrically: The function f(x,y) gives the slope of the tangent line to thesolution curves of the differential equation (1.12.1) at the point (x, y). Thus, bycomputing f(x,y) for various points (x, y), we can draw small line segmentsthrough the point (x, y) with slope f (x,y) to depict how a solution curve wouldpass through (x, y). The resulting picture of line segments is called the slope fieldof the differential equation, and any solution curves to the differential equation inthe xy-plane must be tangent to the slope field at all points.For example, the differential equation dy/dx =−x/y determines a slope fieldconsisting of small line segments that encircle the origin. Indeed, the solutions tothis differential equation consist of concentric circles centered at the origin.One piece of theory is that different solution curves for the same differential equa-tion can never cross (this essentially tells us that an initial-value problem cannothave multiple solutions). Thus, for example, if we find a solution to the differentialequation (1.12.1) of the form y(x) = y0, for some constant y0(recall that such asolution is called an equilibrium solution), then all other solution curves to thedifferential equation must lie entirely above the line y = y0or entirely below it.2. Numerically: Suppose we wish to approximate the solution to the initial-valueproblem (1.12.2) at the point x = x1= x0+h, where h is small. Euler’s methoduses the slope of the solution at (x0,y0), which is f(x0,y0), to use a tangent lineapproximation to the solution:y(x) = y0+ f(x0,y0)(x − x0).Therefore, we approximatey(x1) = y0+ f(x0,y0)(x1− x0) = y0+ hf ( x0,y0).Now, starting from the point (x1,y(x1)), we can repeat the process to find ap-proximations to the solutions at other points x2,x3,.... The conclusion is that theapproximation to the solution to the initial-value problem (1.12.2) at the pointsxn+1= x0+ nh (n = 0, 1,...)isyn+1= yn+ hf ( xn,yn), n = 0, 1,...In Section 1.10, other modifications to Euler’s method are also discussed.“main”2007/2/16page 106iiiiiiii106 CHAPTER 1 First-Order Differential Equations3. Analytically: In some situations, we can explicitly obtain an equation for the gen-eral solution to the differential equation (1.12.1). These include situations in whichthe differential equation is separable, first-order linear, first-order homogeneous,Bernoulli, and/or exact. Table 1.12.1 shows the types of differential equations wecan solve analytically and summarizes the solution techniques. If a given differ-ential equation cannot be written in one of these forms, then the next step is to tryto determine an integrating factor. If that fails, then we might try to find a changeof variables that would reduce the differential equation to one of the above types.Type Standard Form TechniqueSeparable p(y)y= q(x) Separate the variables and integrate.First-order linear y+ p(x)y = q(x) Rewrite asddx(I · y)= I · q(x), whereI = ep(x)dx, and integrate with respect to