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Chapter 11 Boundary Value Problems and Sturm–Liouville Theory 677 11.1 The Occurrence of Two-Point Boundary Value Problems 677
1.3 Classification of Differential Equations
(a) Write down the initial value problem for the filtering process; letq(t)be the amount of dye in the pool at any timet.
(b) Solve the problem in part (a).
(c) You have invited several dozen friends to a pool party that is scheduled to begin in 4 h.
You have also determined that the effect of the dye is imperceptible if its concentration is less than 0.02 g/gal. Is your filtering system capable of reducing the dye concentration to this level within 4 h?
(d) Find the timeTat which the concentration of dye first reaches the value 0.02 g/gal.
(e) Find the flow rate that is sufficient to achieve the concentration 0.02 g/gal within 4 h.
1.3 Classification of Differential Equations
The main purpose of this book is to discuss some of the properties of solutions of differential equations, and to present some of the methods that have proved effective in finding solutions or, in some cases, approximating them. To provide a framework for our presentation, we describe here several useful ways of classifying differential equations.
Ordinary and Partial Differential Equations. One important classification is based on whether the unknown function depends on a single independent variable or on sev- eral independent variables. In the first case, only ordinary derivatives appear in the differential equation, and it is said to be anordinary differential equation. In the sec- ond case, the derivatives are partial derivatives, and the equation is called apartial differential equation.
All the differential equations discussed in the preceding two sections are ordinary differential equations. Another example of an ordinary differential equation is
Ld2Q(t)
dt2 +RdQ(t) dt + 1
C Q(t)=E(t), (1)
for the chargeQ(t)on a capacitor in a circuit with capacitanceC, resistanceR, and inductance L; this equation is derived in Section 3.7. Typical examples of partial differential equations are the heat conduction equation
α2∂2u(x,t)
∂x2 = ∂u(x,t)
∂t (2)
and the wave equation
a2∂2u(x,t)
∂x2 = ∂2u(x,t)
∂t2 . (3)
Here,α2anda2are certain physical constants. Note that in both Eqs. (2) and (3) the dependent variableudepends on the two independent variablesxandt. The heat conduction equation describes the conduction of heat in a solid body, and the wave equation arises in a variety of problems involving wave motion in solids or fluids.
Systems of Differential Equations. Another classification of differential equations de- pends on the number of unknown functions that are involved. If there is a single
20 Chapter 1. Introduction
function to be determined, then one equation is sufficient. However, if there are two or more unknown functions, then a system of equations is required. For example, the Lotka–Volterra, or predator–prey, equations are important in ecological modeling.They have the form
dx/dt=ax−αxy
dy/dt=−cy+γxy, (4)
wherex(t)andy(t)are the respective populations of the prey and predator species.
The constantsa,α,c, andγ are based on empirical observations and depend on the particular species being studied. Systems of equations are discussed in Chapters 7 and 9; in particular, the Lotka–Volterra equations are examined in Section 9.5. In some areas of application it is not unusual to encounter very large systems containing hundreds, or even many thousands, of equations.
Order. Theorderof a differential equation is the order of the highest derivative that appears in the equation. The equations in the preceding sections are all first order equations, whereas Eq. (1) is a second order equation. Equations (2) and (3) are second order partial differential equations. More generally, the equation
F[t,u(t),u′(t),. . .,u(n)(t)] =0 (5) is an ordinary differential equation of thenth order. Equation (5) expresses a relation between the independent variablet and the values of the functionuand its firstn derivativesu′,u′′,. . .,u(n). It is convenient and customary in differential equations to writeyforu(t), withy′,y′′,. . .,y(n)standing foru′(t),u′′(t),. . .,u(n)(t). Thus Eq. (5) is written as
F(t,y,y′,. . .,y(n))=0. (6)
For example,
y′′′+2ety′′+yy′ =t4 (7) is a third order differential equation fory=u(t). Occasionally, other letters will be used instead oft andyfor the independent and dependent variables; the meaning should be clear from the context.
We assume that it is always possible to solve a given ordinary differential equation for the highest derivative, obtaining
y(n) = f(t,y,y′,y′′,. . .,y(n−1)). (8) This is mainly to avoid the ambiguity that may arise because a single equation of the form (6) may correspond to several equations of the form (8). For example, the equation
(y′)2+ty′+4y=0 (9)
leads to the two equations y′= −t+!
t2−16y
2 or y′ = −t−!
t2−16y
2 . (10)
Linear and Nonlinear Equations. A crucial classification of differential equations is whether they are linear or nonlinear. The ordinary differential equation
F(t,y,y′,. . .,y(n))=0
1.3 Classification of Differential Equations 21
is said to be linearifF is a linear function of the variables y,y′,. . .,y(n); a similar definition applies to partial differential equations. Thus the general linear ordinary differential equation of ordernisa0(t)y(n)+a1(t)y(n−1)+ · · · +an(t)y=g(t). (11) Most of the equations you have seen thus far in this book are linear; examples are the equations in Sections 1.1 and 1.2 describing the falling object and the field mouse population. Similarly, in this section, Eq. (1) is a linear ordinary differential equation and Eqs. (2) and (3) are linear partial differential equations. An equation that is not of the form (11) is anonlinear equation. Equation (7) is nonlinear because of the termyy′. Similarly, each equation in the system (4) is nonlinear because of the terms that involve the productxy.
A simple physical problem that leads to a nonlinear differential equation is the oscillating pendulum. The angleθ that an oscillating pendulum of lengthL makes with the vertical direction (see Figure 1.3.1) satisfies the equation
d2θ dt2 + g
L sinθ=0, (12)
whose derivation is outlined in Problems 29 through 31. The presence of the term involving sinθmakes Eq. (12) nonlinear.
L
m
mg θ
FIGURE 1.3.1 An oscillating pendulum.
The mathematical theory and methods for solving linear equations are highly developed. In contrast, for nonlinear equations the theory is more complicated, and methods of solution are less satisfactory. In view of this, it is fortunate that many significant problems lead to linear ordinary differential equations or can be approx- imated by linear equations. For example, for the pendulum, if the angleθ is small, then sinθ∼=θand Eq. (12) can be approximated by the linear equation
d2θ dt2 + g
Lθ=0. (13)
This process of approximating a nonlinear equation by a linear one is calledlineariza- tion; it is an extremely valuable way to deal with nonlinear equations. Nevertheless, there are many physical phenomena that simply cannot be represented adequately by linear equations. To study these phenomena, it is essential to deal with nonlinear equations.
22 Chapter 1. Introduction
In an elementary text it is natural to emphasize the simpler and more straight- forward parts of the subject. Therefore, the greater part of this book is devoted to linear equations and various methods for solving them. However, Chapters 8 and 9, as well as parts of Chapter 2, are concerned with nonlinear equations. Whenever it is appropriate, we point out why nonlinear equations are, in general, more difficult and why many of the techniques that are useful in solving linear equations cannot be applied to nonlinear equations.Solutions. A solution of the ordinary differential equation (8) on the interval α<t<βis a functionφsuch thatφ′,φ′′,. . .,φ(n)exist and satisfy
φ(n)(t)=f[t,φ(t),φ′(t),. . .,φ(n−1)(t)] (14) for every t in α<t <β. Unless stated otherwise, we assume that the function f of Eq. (8) is a real-valued function, and we are interested in obtaining real-valued solutionsy=φ(t).
Recall that in Section 1.2 we found solutions of certain equations by a process of direct integration. For instance, we found that the equation
dp
dt =0.5p−450 (15)
has the solution
p=900+cet/2, (16)
wherecis an arbitrary constant. It is often not so easy to find solutions of differential equations. However, if you find a function that you think may be a solution of a given equation, it is usually relatively easy to determine whether the function is actually a solution simply by substituting the function into the equation. For example, in this way it is easy to show that the functiony1(t)=costis a solution of
y′′+y=0 (17)
for allt. To confirm this, observe thaty′1(t)=−sintandy′′1(t)=−cost; then it follows thaty′′1(t)+y1(t)=0. In the same way you can easily show thaty2(t)=sint is also a solution of Eq. (17). Of course, this does not constitute a satisfactory way to solve most differential equations, because there are far too many possible functions for you to have a good chance of finding the correct one by a random choice. Nevertheless, you should realize that you can verify whether any proposed solution is correct by substituting it into the differential equation. This can be a very useful check; it is one that you should make a habit of considering.
Some Important Questions. Although for the equations (15) and (17) we are able to verify that certain simple functions are solutions, in general we do not have such solutions readily available. Thus a fundamental question is the following: Does an equation of the form (8) always have a solution? The answer is “No.” Merely writing down an equation of the form (8) does not necessarily mean that there is a function y=φ(t)that satisfies it. So, how can we tell whether some particular equation has a solution?This is the question ofexistenceof a solution,and it is answered by theorems stating that under certain restrictions on the functionf in Eq. (8), the equation always has solutions. This is not a purely mathematical concern for at least two reasons.