1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 Definitions and Terminology 1.2 Initial-Value Problems 1.3 Differential Equations as Mathematical Models CHAPTER 1 IN REVIEW The words differential and equations certainly suggest solving some kind of equation that contains derivatives y',y".....Analogous to a course in algebra and trigonometry,in which a good amount of time is spent solving equations such as x2+5x+4=0 for the unknown number x,in this course one of our tasks will be to solve differential equations such as y"+2y'+y=0 for an unknown function y=中(x). The preceding paragraph tells something,but not the complete story,about the course you are about to begin.As the course unfolds,you will see that there is more to the study of differential equations than just mastering methods that someone has devised to solve them. But first things first.In order to read,study,and be conversant in a specialized subject,you have to learn the terminology of that discipline.This is the thrust of the first two sections of this chapter.In the last section we briefly examine the link between differential equations and the real world.Practical questions such as How fast does a disease spread?How fast does a population change?involve rates of change or derivatives.As so the mathematical description-or mathematical model-of experiments,observations,or theories may be a differential equation. 1
1 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 Definitions and Terminology 1.2 Initial-Value Problems 1.3 Differential Equations as Mathematical Models CHAPTER 1 IN REVIEW The words differential and equations certainly suggest solving some kind of equation that contains derivatives y, y , . . . . Analogous to a course in algebra and trigonometry, in which a good amount of time is spent solving equations such as x2 5x 4 0 for the unknown number x, in this course one of our tasks will be to solve differential equations such as y 2y y 0 for an unknown function y (x). The preceding paragraph tells something, but not the complete story, about the course you are about to begin. As the course unfolds, you will see that there is more to the study of differential equations than just mastering methods that someone has devised to solve them. But first things first. In order to read, study, and be conversant in a specialized subject, you have to learn the terminology of that discipline. This is the thrust of the first two sections of this chapter. In the last section we briefly examine the link between differential equations and the real world. Practical questions such as How fast does a disease spread? How fast does a population change? involve rates of change or derivatives. As so the mathematical description—or mathematical model—of experiments, observations, or theories may be a differential equation.
CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 DEFINITIONS AND TERMINOLOGY REVIEW MATERIAL Definition of the derivative Rules of differentiation Derivative as a rate of change First derivative and increasing/decreasing Second derivative and concavity INTRODUCTION The derivative dy/dx of a function y=(x)is itself another function '(x) found by an appropriate rule.The function y=eis differentiable on the interval (,)and by the Chain Rule its derivative is dy/dx =0.2xe If we replaceon the right-hand side of the last equation by the symbol y,the derivative becomes dy dx =0.2y. (1) Now imagine that a friend of yours simply hands you equation(1)-you have no idea how it was constructed-and asks,What is the function represented by the symbol y?You are now face to face with one of the basic problems in this course: How do you solve such an equation for the unknown function y=b(x)? A DEFINITION The equation that we made up in (1)is called a differential equation.Before proceeding any further,let us consider a more precise definition of this concept. DEFINITION 1.1.1 Differential Equation An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables,is said to be a differential equation (DE). To talk about them,we shall classify differential equations by type,order,and linearity. CLASSIFICATION BY TYPE If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable it is said to be an ordinary differential equation (ODE).For example, A DE can contain more than one dependent variable dy 5y=e". d dy_y+6y=0, d dx and d+y=2x+y(② dt dt are ordinary differential equations.An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a
DEFINITIONS AND TERMINOLOGY REVIEW MATERIAL ● Definition of the derivative ● Rules of differentiation ● Derivative as a rate of change ● First derivative and increasing/decreasing ● Second derivative and concavity INTRODUCTION The derivative dydx of a function y (x) is itself another function (x) found by an appropriate rule. The function is differentiable on the interval ( , ), and by the Chain Rule its derivative is . If we replace on the right-hand side of the last equation by the symbol y, the derivative becomes . (1) Now imagine that a friend of yours simply hands you equation (1)—you have no idea how it was constructed—and asks, What is the function represented by the symbol y? You are now face to face with one of the basic problems in this course: How do you solve such an equation for the unknown function y (x)? dy dx 0.2xy e0.1x2 dy>dx 0.2xe0.1x2 y e0.1x2 2 ● CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1.1 A DEFINITION The equation that we made up in (1) is called a differential equation. Before proceeding any further, let us consider a more precise definition of this concept. DEFINITION 1.1.1 Differential Equation An equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables, is said to be a differential equation (DE). To talk about them, we shall classify differential equations by type, order, and linearity. CLASSIFICATION BY TYPE If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable it is said to be an ordinary differential equation (ODE). For example, A DE can contain more than one dependent variable (2) are ordinary differential equations. An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a dy dx 5y ex , d2y dx2 dy dx 6y 0, and dx dt dy dt 2x y b b
1.1 DEFINITIONS AND TERMINOLOGY 。3 partial differential equation (PDE).For example. 2u Bu + 0 票染贵 and ou=_dv at (3) dy ax are partial differential equations." Throughout this text ordinary derivatives will be written by using either the Leibniz notation dy/dx,d2y/dx2,d3y/dx3,...or the prime notation y',y",y",.... By using the latter notation,the first two differential equations in(2)can be written a little more compactly as y'+5y =et and y"-y'+6y =0.Actually,the prime notation is used to denote only the first three derivatives;the fourth derivative is written y(4)instead of y".In general,the nth derivative ofy is written d"y/dx"or y(m). Although less convenient to write and to typeset,the Leibniz notation has an advan- tage over the prime notation in that it clearly displays both the dependent and independent variables.For example,in the equation unknown function ordependent variable d2x d2 +16r=0 Lindependent variable it is immediately seen that the symbol x now represents a dependent variable. whereas the independent variable is t.You should also be aware that in physical sciences and engineering,Newton's dot notation (derogatively referred to by some as the "flyspeck"notation)is sometimes used to denote derivatives with respect to time t.Thus the differential equation d2s/dt2=-32 becomes $=-32.Partial derivatives are often denoted by a subscript notation indicating the indepen- dent variables.For example,with the subscript notation the second equation in (3)becomes u=un-2ut. CLASSIFICATION BY ORDER The order of a differential equation (either ODE or PDE)is the order of the highest derivative in the equation.For example, second order- 厂first order .+5 3 -4y =ex dx is a second-order ordinary differential equation.First-order ordinary differential equations are occasionally written in differential form M(x,y)dx N(x,y)dy =0. For example,if we assume that y denotes the dependent variable in (y-x)dx 4x dy =0,then y'=dy/dx,so by dividing by the differential dx,we get the alternative form 4xy'+y =x.See the Remarks at the end of this section. In symbols we can express an nth-order ordinary differential equation in one dependent variable by the general form Fx,,y',,ym)=0, (4) where Fis a real-valued function of n+2 variables:x,y,y',...,y(m).For both prac- tical and theoretical reasons we shall also make the assumption hereafter that it is possible to solve an ordinary differential equation in the form(4)uniquely for the Except for this introductory section,only ordinary differential equations are considered in A Firsr Course in Differential Equations with Modeling Applications.Ninth Edition.In that text the word eguation and the abbreviation DE refer only to ODEs.Partial differential equations or PDEs are considered in the expanded volume Differential Equations with Boundary-Value Problems, Seventh Edition
partial differential equation (PDE). For example, (3) are partial differential equations.* Throughout this text ordinary derivatives will be written by using either the Leibniz notation dydx, d2ydx2 , d3ydx3 , . . . or the prime notation y, y , y ,.... By using the latter notation, the first two differential equations in (2) can be written a little more compactly as y 5y ex and y y 6y 0. Actually, the prime notation is used to denote only the first three derivatives; the fourth derivative is written y(4) instead of y . In general, the nth derivative of y is written dnydxn or y(n) . Although less convenient to write and to typeset, the Leibniz notation has an advantage over the prime notation in that it clearly displays both the dependent and independent variables. For example, in the equation it is immediately seen that the symbol x now represents a dependent variable, whereas the independent variable is t. You should also be aware that in physical sciences and engineering, Newton’s dot notation (derogatively referred to by some as the “flyspeck” notation) is sometimes used to denote derivatives with respect to time t. Thus the differential equation d2 sdt2 32 becomes s¨ 32. Partial derivatives are often denoted by a subscript notation indicating the independent variables. For example, with the subscript notation the second equation in (3) becomes uxx utt 2ut. CLASSIFICATION BY ORDER The order of a differential equation (either ODE or PDE) is the order of the highest derivative in the equation. For example, is a second-order ordinary differential equation. First-order ordinary differential equations are occasionally written in differential form M(x, y) dx N(x, y) dy 0. For example, if we assume that y denotes the dependent variable in (y x) dx 4x dy 0, then y dydx, so by dividing by the differential dx, we get the alternative form 4xy y x. See the Remarks at the end of this section. In symbols we can express an nth-order ordinary differential equation in one dependent variable by the general form , (4) where F is a real-valued function of n 2 variables: x, y, y, . . . , y(n) . For both practical and theoretical reasons we shall also make the assumption hereafter that it is possible to solve an ordinary differential equation in the form (4) uniquely for the F(x, y, y, . . . , y(n) ) 0 second order first order 5( )3 4y ex dy ––– dx d2y –––– dx2 d2x ––– dt2 16x 0 unknown function or dependent variable independent variable 2 u x2 2 u y2 0, 2 u x2 2 u t 2 2 u t , and u y v x 1.1 DEFINITIONS AND TERMINOLOGY ● 3 * Except for this introductory section, only ordinary differential equations are considered in A First Course in Differential Equations with Modeling Applications, Ninth Edition. In that text the word equation and the abbreviation DE refer only to ODEs. Partial differential equations or PDEs are considered in the expanded volume Differential Equations with Boundary-Value Problems, Seventh Edition.
4.CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS highest derivative y()in terms of the remaining n+1 variables.The differential equation d"y dx" =f(r,y.y,....y-D) (5) where fis a real-valued continuous function,is referred to as the normal form of(4). Thus when it suits our purposes,we shall use the normal forms dy d-y f(x.y) and f(x,y,y') dx to represent general first-and second-order ordinary differential equations.For example, the normal form of the first-order equation 4xy'+y =x is y'=(x-y)/4x;the normal form of the second-order equationy"-y'+6y =0 is y"=y'-6y.See the Remarks. CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation(4) is said to be linear if F is linear iny,y'.....m.This means that an nth-order ODE is linear when (4)is an()y(m)an-(x)y(-1)+...+a(x)y'ao(x)y-g(x)=0 or d"y dm-ly a(x) an-1(x) +…+a ao(x)y =g(x). (6) dxn dx Two important special cases of(6)are linear first-order(n=1)and linear second- order (n=2)DEs: dy a(x) +do(x)y =g(x)and +a, d2y a2(x) dx +ay=8.(⑦ dx In the additive combination on the left-hand side of equation (6)we see that the char- acteristic two properties of a linear ODE are as follows: The dependent variable y and all its derivatives y',y",....y(m are of the first degree,that is,the power of each term involving y is 1. The coefficients ao,a.....an of y,y',....y(m)depend at most on the independent variable x. The equations (y -x)dx 4x dy =0.y"-2y'+y=0.and dy dy_5y=e dr3+x dx are,in turn,linear first-,second-,and third-order ordinary differential equations.We have just demonstrated that the first equation is linear in the variable y by writing it in the alternative form 4xy'+y x.A nonlinear ordinary differential equation is sim- ply one that is not linear.Nonlinear functions of the dependent variable or its deriva- tives,such as sin y or e,cannot appear in a linear equation.Therefore nonlinear term: nonlinear term: nonlinear term: coefficient depends on y nonlinear function of y power not 1 d-y (1-y)y'+2y=e, +siny=0 d and d+y2=0 are examples of nonlinear first-,second-,and fourth-order ordinary differential equa- tions,respectively. SOLUTIONS As was stated before,one of the goals in this course is to solve,or find solutions of,differential equations.In the next definition we consider the con- cept of a solution of an ordinary differential equation
highest derivative y(n) in terms of the remaining n 1 variables. The differential equation , (5) where f is a real-valued continuous function, is referred to as the normal form of (4). Thus when it suits our purposes, we shall use the normal forms to represent general first- and second-order ordinary differential equations. For example, the normal form of the first-order equation 4xy y x is y (x y)4x; the normal form of the second-order equation y y 6y 0 is y y 6y. See the Remarks. CLASSIFICATION BY LINEARITY An nth-order ordinary differential equation (4) is said to be linear if F is linear in y, y,..., y(n) . This means that an nth-order ODE is linear when (4) is an(x)y(n) an1(x)y(n1) a1(x)y a0(x)y g(x) 0 or . (6) Two important special cases of (6) are linear first-order (n 1) and linear secondorder (n 2) DEs: . (7) In the additive combination on the left-hand side of equation (6) we see that the characteristic two properties of a linear ODE are as follows: • The dependent variable y and all its derivatives y, y , . . . , y(n) are of the first degree, that is, the power of each term involving y is 1. • The coefficients a0, a1, . . . , an of y, y, . . . , y(n) depend at most on the independent variable x. The equations are, in turn, linear first-, second-, and third-order ordinary differential equations. We have just demonstrated that the first equation is linear in the variable y by writing it in the alternative form 4xy y x. A nonlinear ordinary differential equation is simply one that is not linear. Nonlinear functions of the dependent variable or its derivatives, such as sin y or , cannot appear in a linear equation. Therefore are examples of nonlinear first-, second-, and fourth-order ordinary differential equations, respectively. SOLUTIONS As was stated before, one of the goals in this course is to solve, or find solutions of, differential equations. In the next definition we consider the concept of a solution of an ordinary differential equation. nonlinear term: coefficient depends on y nonlinear term: nonlinear function of y nonlinear term: power not 1 (1 y)y 2y ex , sin y 0, and d2y –––– dx2 y 2 0 d4y –––– dx 4 ey (y x)dx 4x dy 0, y 2y y 0, and d3 y dx3 x dy dx 5y ex a1(x) dy dx a0(x)y g(x) and a2(x) d2 y dx2 a1(x) dy dx a0(x)y g(x) an(x) dny dxn an1(x) dn1 y dxn1 a1(x) dy dx a0(x)y g(x) dy dx f(x, y) and d2y dx2 f(x, y, y) dny dxn f(x, y, y, . . . , y(n1)) 4 ● CHAPTER 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1 DEFINITIONS AND TERMINOLOGY 5 DEFINITION 1.1.2 Solution of an ODE Any function b,defined on an interval and possessing at least n derivatives that are continuous on /which when substituted into an nth-order ordinary differential equation reduces the equation to an identity,is said to be a solution of the equation on the interval. In other words,a solution of an nth-order ordinary differential equation(4)is a func- tion that possesses at least n derivatives and for which Fx,中x),b'(x),..,中mx)=0 for all x in/. We say that satisfies the differential equation on I.For our purposes we shall also assume that a solution o is a real-valued function.In our introductory discussion we saw that y=eis a solution of dy/dx =0.2xy on the interval (,) Occasionally,it will be convenient to denote a solution by the alternative symbol y(x). INTERVAL OF DEFINITION You cannot think solution of an ordinary differential equation without simultaneously thinking interval.The interval in Definition 1.1.2 is variously called the interval of definition,the interval of existence,the interval of validity,or the domain of the solution and can be an open interval(a,b),a closed interval [a,b],an infinite interval (a.)and so on. EXAMPLE 1 Verification of a Solution Verify that the indicated function is a solution of the given differential equation on the interval (-o,) (a)dy/dx=xyi2;y=x (b)y"-2y'+y=0;y=xe SOLUTION One way of verifying that the given function is a solution is to see,after substituting,whether each side of the equation is the same for every x in the interval. (a)From dy left-hand side: (4·x=x3 dx 16 1/2 right-hand side: xy2=x· we see that each side of the equation is the same for every real number x.Note that y=x2is,by definition,the nonnegative square root of (b)From the derivatives y'=xe+e'and y"=xet+2e*we have,for every real numberx, left-hand side: y"-2y'+y=(xe+2e)-2xe+e)+xex=0, right-hand side: 0. ■ Note,too,that in Example I each differential equation possesses the constant so- lution y=0,<x<.A solution of a differential equation that is identically zero on an interval is said to be a trivial solution. SOLUTION CURVE The graph of a solution of an ODE is called a solution curve.Since is a differentiable function,it is continuous on its interval of defini- tion.Thus there may be a difference between the graph of the function and the
right-hand side: 0. left-hand side: y 2y y (xex 2ex ) 2(xex ex ) xex 0, 1 16 x4 y1/2 1 4 x2 right-hand side: xy1/2 x 1 16 x4 1/2 x 1 4 x2 1 4 x3 , left-hand side: dy dx 1 16 (4 x3 ) 1 4 x3 , y 2y y 0; y xex dy>dx xy1/2; y 1 16 x4 y e0.1x2 F(x, (x), (x), . . . , (n) (x)) 0 for all x in I. 1.1 DEFINITIONS AND TERMINOLOGY ● 5 DEFINITION 1.1.2 Solution of an ODE Any function , defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into an nth-order ordinary differential equation reduces the equation to an identity, is said to be a solution of the equation on the interval. In other words, a solution of an nth-order ordinary differential equation (4) is a function that possesses at least n derivatives and for which We say that satisfies the differential equation on I. For our purposes we shall also assume that a solution is a real-valued function. In our introductory discussion we saw that is a solution of dydx 0.2xy on the interval (, ). Occasionally, it will be convenient to denote a solution by the alternative symbol y(x). INTERVAL OF DEFINITION You cannot think solution of an ordinary differential equation without simultaneously thinking interval. The interval I in Definition 1.1.2 is variously called the interval of definition, the interval of existence, the interval of validity, or the domain of the solution and can be an open interval (a, b), a closed interval [a, b], an infinite interval (a, ), and so on. EXAMPLE 1 Verification of a Solution Verify that the indicated function is a solution of the given differential equation on the interval (, ). (a) (b) SOLUTION One way of verifying that the given function is a solution is to see, after substituting, whether each side of the equation is the same for every x in the interval. (a) From we see that each side of the equation is the same for every real number x. Note that is, by definition, the nonnegative square root of . (b) From the derivatives y xex ex and y xex 2ex we have, for every real number x, Note, too, that in Example 1 each differential equation possesses the constant solution y 0, x . A solution of a differential equation that is identically zero on an interval I is said to be a trivial solution. SOLUTION CURVE The graph of a solution of an ODE is called a solution curve. Since is a differentiable function, it is continuous on its interval I of definition. Thus there may be a difference between the graph of the function and the