Mathematics
Differential Equations
Differential equations are mathematical equations that describe how a quantity changes as a function of its own value and other variables. They are used to model a wide range of phenomena in science and engineering, from population growth to the behavior of electrical circuits. Solving differential equations involves finding a function that satisfies the equation and often requires advanced mathematical techniques.
Written by Perlego with AI-assistance
Related key terms
1 of 5
12 Key excerpts on "Differential Equations"
eBook - PDF- Virginia W. Noonburg(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
1 Introduction to Differential Equations Differential Equations arise from real-world problems and problems in applied math-ematics. One of the first things you are taught in calculus is that the derivative of a function is the instantaneous rate of change of the function with respect to its inde-pendent variable. When mathematics is applied to real-world problems, it is often the case that finding a relation between a function and its rate of change is easier than find-ing a formula for the function itself; it is this relation between an unknown function and its derivatives that produces a differential equation. To give a very simple example, a biologist studying the growth of a population, with size at time ? given by the function ?(?) , might make the very simple, but logi-cal, assumption that a population grows at a rate directly proportional to its size. In mathematical notation, the equation for ?(?) could then be written as ?? ?? = ??(?), where the constant of proportionality, ? , would probably be determined experimentally by biologists working in the field. Equations used for modeling population growth can be much more complicated than this, sometimes involving scores of interacting populations with different properties; however, almost any population model is based on equations similar to this. In an analogous manner, a physicist might argue that all the forces acting on a particular moving body at time ? depend only on its position ?(?) and its velocity ? ′ (?) . He could then use Newton’s second law to express mass times acceleration as ?? ″ (?) and write an equation for ?(?) in the form ?? ″ (?) = ?(?(?), ? ′ (?)), where ? is some function of two variables. One of the best-known equations of this type is the spring-mass equation ?? ″ + ?? ′ + ?? = ?(?), (1.1) 1
- Yuefan Deng(Author)
- 2014(Publication Date)
- World Scientific(Publisher)
1 Chapter 1 First-Order Differential Equations 1.1 Definition of Differential Equations A differential equation (DE) is a mathematical equation that relates some functions of one or more variables with its derivatives. A DE is used to describe changing quantities and it plays a major role in quantitative studies in many disciplines such as all areas of engineering, physical sciences, life sciences, and economics. Examples Are they DEs or not? + + = 0 No Chapter 1 First-Order Differential Equations 2 + ′ + = 0 Yes Here ′ = + ′ + ′ = 0 Yes Here ′ = and ′ = ′′ = Yes Here ′ = To solve a DE is to express the solution of the unknown function (the dependent variable) in mathematical terms without the derivatives. Example + = 0 ′ = − is not a solution = − is a solution In general, there are two common ways in solving DEs, analytic and numerical. Most DEs, difficult to solve by analytical methods, must be “solved” by numerical methods although many DEs are too stiff to solve using numerical techniques. Solving DEs by numerical methods is a different subject requiring basic knowledge of computer programming and numerical analysis; this book focuses on introducing analytical methods for solving very small families of DEs that are truly solvable. Although the DEs are quite simple, the solution methods are not and the essential solution steps and terminologies involved are fully applicable to much more complicated DEs. Classification of DEs Classification of DEs is itself another subject in studying DEs. We will introduce classifications and terminologies for flowing the contents of the book but one may still need to lookup terms undefined here.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 2 Types of Differential Equations 1. Differential equation Visualization of heat transfer in a pump casing, by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential Equations play a prominent role in engineering, physics, economics, and other disciplines. ________________________ WORLD TECHNOLOGIES ________________________ Differential Equations arise in many areas of science and technology, specifically when-ever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. An example of modelling a real world problem using Differential Equations is deter-mination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This means the ball's acceleration, which is the derivative of its velocity, depends on the velocity.
- Albert L. Rabenstein(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
Introduction to Differential Equations 1.1 INTRODUCTION An ordinary differential equation may be defined as an equation that in-volves a single unknown function of a single variable and some finite number of its derivatives. For example, a simple problem from calculus is that of finding all functions/for which /'(*) = 3x 2 -4x + 5 (1.1) for all x. Clearly a function/satisfies the condition (1.1) if and only if it is of the form f(x) = x 3 - 2x 2 + 5x + c , where c is an arbitrary number. A more difficult problem is that of finding all functions g for which gx) + 2[0(x)] 2 = 3x 2 - 4x + 5 . (1.2) Another difficult problem is that of finding all functions y for which (we use the abbreviation y for y(x)) d 2i dx 2 , i d y 2 A — 3x1 — 1 +4y = sinx. / (1.3) 2 Introduction to Differential Equations In each of the problems (1.1), (1.2), and (1.3) we are asked to find all functions that satisfy a certain condition, where the condition involves one or more derivatives of the function. We can reformulate our definition of a differential equation as follows. Let F be a function of n + 2 variables. Then the equation Fix,y, / , / , . . . , / w ) ] = 0 (1.4) is called an ordinary differential equation of order n for the unknown func-tion y. The order of the equation is the order of the highest order derivative that appears in the equation. Thus, Eqs. (1.1) and (1.2) are first-order equa-tions, while Eq. (1.3) is of second order. A partial differential equation (as distinguished from an ordinary differen-tial equation) is an equation that involves an unknown function of more than one independent variable, together with partial derivatives of the function. An example of a partial differential equation for an unknown function u(x, t) of two variables is d 2 u du Almost all the Differential Equations that we shall consider will be ordinary.
eBook - PDF- Dennis Zill(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Kevin George/Shutterstock.com 2 1.1 Definitions and Terminology 1.2 Initial-Value Problems 1.3 Differential Equations as Mathematical Models C H A P T E R 1 I N R E V I E W T he words differential and equations suggest solving some kind of equation that contains derivatives y 9 , y 0 , Á . Analogous to a course in algebra, in which a good amount of time is spent solving equations such as x 2 1 5 x 1 4 5 0 for the unknown number x , in this course one of our tasks will be to solve Differential Equations such as y 0 1 2 y 9 1 y 5 0 for an unknown function y 5 f ( x ). As the course unfolds, you will see there is more to the study of Differential Equations than just mastering methods that mathematicians over past centuries devised to solve them. But first things first. In order to read, study, and be conversant in a specialized subject you have to learn some of 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. 1 Introduction to Differential Equations Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 INTRODUCTION The derivative dy y dx of a function y 5 f ( x ) is itself another function f 9 ( x ) found by an appropriate rule. The exponential function y 5 e 0.1 x 2 is differentiable on the interval ( 2` , ` ) and by the Chain Rule its first derivative is dy y dx 5 0.2 xe 0.1 x 2 . If we replace e 0.1 x 2 on the right-hand side of the last equation by the symbol y , the derivative becomes dy dx 5 0.2 xy .
No longer available |Learn moreEngineering Mathematics Exam Prep
Problems and Solutions
- A. Saha, D. Dutta, S. Kar, P. Majumder, A. Paul, S. Musa, PhD(Authors)
- 2023(Publication Date)
- Mercury Learning and Information(Publisher)
C H A P T E R 4 ORDINARY Differential Equations 4.1 BASIC CONCEPTS 4.1.1 Definition of a Differential Equation A differential equation is an equation involving at least one differential or differential coefficient with or without variables. Examples: (i) dy = (2x – e y )dx In this equation the differen- tials are dx and dy; dependent variable is “y” and independent variable is “x.” (ii) 3 3 2 3 2 4 x d y d y e dx dx æ ö + = ç ÷ ç ÷ è ø In this equation, the differential coefficients are 3 3 d y dx and 2 2 d y dx ; dependent variables is “y” and independent variable is “x.” (iii) 2 2 0 d y dx = In this equation, the only differential coeffi- cient is 2 2 d y dx ; dependent variables is “y” and independent variable is “x.” (iv) 2 2 0 y u u x e x y ¶ ¶ + = ¶ ¶ In this above equation, differential coefficients are u x ¶ ¶ and 2 2 u y ¶ ¶ ; “u” is the dependent variable, where as “x” and “y” are independent vari- ables. (v) 2 2 2 2 2 2 2 3 4 0 u u u x y z ¶ ¶ ¶ + + = ¶ ¶ ¶ In this above equation, differential coefficients are 2 2 u x ¶ ¶ , 2 2 u y ¶ ¶ and 2 2 u z ¶ ¶ ; dependent variable is “u” and independent variables are “x, ” “y,” and “z.” 4.1.2 Classification of Differential Equations (a) Ordinary differential equation: An ordinary differential equation is an equation in- volving derivatives or differentials with respect to a single independent variable. Examples: (i) dy = (2x – e y )dx (ii) 2 2 2 3 d y dy x x dx dx + = (b) Partial differential equation: A partial differential equation is an equation in- volving partial derivatives or differentials w.r.t at least two independent variables. Examples: (i) 0 u u x y x y ¶ ¶ + = ¶ ¶ (ii) 2 2 2 2 2 2 2 0 u u u xy z y x y z ¶ ¶ ¶ + + = ¶ ¶ ¶ 4.1.3 Order of a Differential Equation The order of a differential equation is the order of the highest derivative or highest differential occur- ring in the equation.
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
14 Ordinary Differential Equations Differential Equations are the group of equations that contain derivatives. There are several different types of Differential Equations, but here we will be considering only the simplest types. As its name suggests, an ordinary differential equation (ODE) contains only ordinary derivatives (no partial derivatives) and describes the relationship between these derivatives of the dependent variable , usually called y , with respect to the independent variable , usually called x . The solution to such an ODE is therefore a function of x and is written y ( x ). For an ODE to have a closed-form solution, it must be possible to express y ( x ) in terms of the standard elementary functions such as x 2 , √ x , exp x , ln x , sin x , etc. The solutions of some Differential Equations cannot, however, be written in closed form, but only as an infinite series that carry no special names. Ordinary Differential Equations may be separated conveniently into different categories according to their general characteristics. The primary grouping adopted here is by the order of the equation. The order of an ODE is simply the order of the highest derivative it contains. Thus, equations containing dy / dx , but no higher derivatives, are called first order, those containing d 2 y / dx 2 are called second order and so on. In this chapter we consider first-order equations and some of the more straightforward equations of second order. Ordinary Differential Equations may be classified further according to degree . The degree of an ODE is the power to which the highest order derivative is raised, after the equation has been rationalised to contain only integer powers of derivatives. Hence the ODE d 3 y dx 3 + x dy dx 3 / 2 + x 2 y = 0 is of third order and second degree, since after rationalisation it contains the term ( d 3 y / dx 3 ) 2 .
eBook - PDF- G Gandolfo(Author)
- 1971(Publication Date)
- North Holland(Publisher)
PART II Differential Equations (LINEAR AND WITH CONSTANT COEFFICIENTS) This page intentionally left blank 1 General Principles An ordinary differential equation is a functional equation involving one or more of the derivatives yyy etc., of an unknown function of time y = /(f), which obviously is differentiable. We have called the equation an ordinary differential equation since the unknown function is a function of only one argument; if the independent variables were more than one, partial derivatives would appear in the differential equation, which would be a partial differential equation (we shall not treat this type of func-tional equation). The order of a differential equation is given by the highest derivative ap-pearing in the equation. After what we said in the Introduction, it will be clear that to solve (or to 'integrate') a differential equation means to find the unknown function that satisfies the relationship expressed by the equation. Let us begin, as usual, by a simple example. Consider the differential equation y = a, where a is a constant. From elementary integral calculus it follows that y = at + b; let us remember, incidentally, that integration — the 'inverse' operation of differentiation — represents the solution of a differen-tial equation. We note that in the solution an arbitrary constant b appears (the other constant a is known, since it appears in the differential equation). 169 170 GENERAL PRINCIPLES Ch. 1 This is not surprising, since we know that differentiation eliminates such a constant, so that from y = at + b we obtain, in fact, y = a. Let us now con-sider the second-order differential equation y = a. Performing two successive integrations we obtain y - at 2 + bt + c; now, two arbitrary constants, b and c, appear in the solution.
eBook - PDFMathematical Methods for Physicists
A Concise Introduction
- Tai L. Chow(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
2 Ordinary dierential equations Physicists have a variety of reasons for studying dierential equations: almost all the elementary and numerous of the advanced parts of theoretical physics are posed mathematically in terms of dierential equations. We devote three chapters to dierential equations. This chapter will be limited to ordinary dierential equations that are reducible to a linear form. Partial dierential equations and special functions of mathematical physics will be dealt with in Chapters 10 and 7. A dierential equation is an equation that contains derivatives of an unknown function which expresses the relationship we seek. If there is only one independent variable and, as a consequence, total derivatives like dx = dt , the equation is called an ordinary dierential equation (ODE). A partial dierential equation (PDE) contains several independent variables and hence partial deriva-tives. The order of a dierential equation is the order of the highest derivative appear-ing in the equation; its degree is the power of the derivative of highest order after the equation has been rationalized, that is, after fractional powers of all deriva-tives have been removed. Thus the equation d 2 y dx 2 3 dy dx 2 y 0 is of second order and first degree, and d 3 y dx 3 1 dy = dx 3 q is of third order and second degree, since it contains the term ( d 3 y = dx 3 2 after it is rationalized. 62 A dierential equation is said to be linear if each term in it is such that the dependent variable or its derivatives occur only once, and only to the first power. Thus d 3 y dx 3 y dy dx 0 is not linear, but x 3 d 3 y dx 3 e x sin x dy dx y ln x is linear. If in a linear dierential equation there are no terms independent of y , the dependent variable, the equation is also said to be homogeneous ; this would have been true for the last equation above if the ‘ln x ’ term on the right hand side had been replaced by zero.
Available until 9 Apr |Learn more- Carlos A. Smith, Scott W. Campbell(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
Thus, the solution of the differential equation describes how the level in the tank changes as a function of time, including its final steady value . The differential equation provides more information than the algebraic equation. 5 Introduction 1.2 Differential Equations Up to this moment in your study of engineering and science you have become quite famil-iar with algebraic equations , for example, x 3 + 3 x 2 + 4 x = 3 t 2 + 4 t (1.15) From this equation we can solve for x at any t . We could repeat this for different values of t and make a graph of x versus t . Many natural and man-made phenomena and systems cannot be described by algebraic equations, or the description provided by algebraic equations is not complete (as in the 0 5 10 15 20 25 30 35 40 50 100 150 200 250 0 5 10 15 20 25 30 35 40 3 3.5 4 4.5 5 5.5 6 Time, min Time, min Flow 2 , kg/min Level h , m FIGURE 1.2 Response of level in tank. 6 A First Course in Differential Equations, Modeling, and Simulation example in Section 1.1). In these cases, Differential Equations (DEs) may provide the required description; an example of a differential equation is d x t dt d x t dt dx t dt x t F t 3 3 2 2 3 4 ( ) ( ) ( ) ( ) ( ) + + + = (1.16) Note that the variable x that is differentiated must be a function of t ; otherwise, the differ-ential would be zero. That is the reason for writing x ( t ). What often happens is that because obviously x is a function of t , we drop the t term and simply write d x dt d x dt dx dt x F t 3 3 2 2 3 4 + + + = ( ) (1.17) A differential equation is an equation containing one or more derivatives of an unknown function and perhaps the function itself . In Equation 1.16, x ( t ), or x in Equation 1.17, is the unknown function; in Equation 1.13, h is the unknown function.
eBook - PDFAn Introduction to Differential Equations
Deterministic Modeling, Methods and Analysis(Volume 1)
- Anil G Ladde, G S Ladde;;;(Authors)
- 2012(Publication Date)
- WSPC(Publisher)
Chapter 2 First-Order Differential Equations 2.1 Introduction In this chapter, mathematical modeling, procedures for solving first-order scalar lin-ear Differential Equations, and their fundamental conceptual analysis are developed. Prior to the presentation of the technical procedures and the concepts, an attempt is made to dispel any doubts or to answer any questions that are frequently asked by students. These include: Why is this course required? Why should I learn this material? How will this help me? The mathematical modeling in Section 2.2 takes a proactive approach to moti-vate the student. Section 2.3 deals with first-order Differential Equations whose solu-tions can be directly found by the methods of integration. This class of first-order Differential Equations is referred to as integrable Differential Equations. Moreover, the mathematical models of laminar blood flow in an artery and the motion of particles in the air are presented to illustrate the usage of this class of differen-tial equations. Section 2.4 is devoted to first-order scalar homogeneous differen-tial equations. The eigenvalue-type method is utilized to solve this class of dif-ferential equations with both constant and variable coefficients. This approach is motivated by observing the fact that the problems of solving linear scalar differ-ential equations are analogous to the problems of solving linear scalar algebraic equations. By integrating the knowledge of the derivatives of exponential func-tions and the concept of solution of a scalar differential equation, the problem of finding a solution to a linear scalar differential equation is reduced to the prob-lem of solving linear algebraic equations. The step-by-step procedures for finding the general solutions and the solutions of initial value problems are logically and clearly outlined. Various examples and illustrations are utilized to better describe the procedures and the usefulness of the Differential Equations.
eBook - PDFA Mathematical Bridge
An Intuitive Journey in Higher Mathematics
- Stephen Hewson(Author)
- 2009(Publication Date)
- WSPC(Publisher)
Chapter 5 Calculus and Differential Equations In 1687 Newton published his monumental work ‘The Mathematical Princi-ples of Natural Philosophy’. In the opening of this great book is a statement of Newton’s second law of motion, which may be paraphrased as follows: force equals mass times acceleration. It is remarkable that a physical law so simple to express would give rise to a branch of mathematics which would prove to be so essential to the further development of most areas of mathe-matics and the natural sciences. This mathematics is the theory of calculus and Differential Equations. In addition to their interest from the point of view of beautiful mathematics, Differential Equations are essential tools in the study of disciplines as diverse as economics and the biological sciences. We begin this chapter by motivating the concepts of differentiation and dif-ferential equations through a discussion of Newton’s second law of motion, rediscovering and extending the ideas of calculus discussed in the study of analysis. 5.1 The Why and How of Calculus Newton’s law of motion has a virtually limitless number of useful applica-tions, from ballistics to planetary dynamics. To obtain an equation for the position of a body in space at each point in time from Newton’s second law we need to probe the relationship between acceleration and position. 5.1.1 Acceleration, velocity and position What is acceleration? If a car travelling at a velocity of v ( t 0 ) miles per hour at a time t 0 smoothly increases its velocity to v ( t 1 ) miles per hour at a time t 1 then the magnitude of its acceleration a over this time period 289 290 A Mathematical Bridge is given by the change in velocity δv = v ( t 1 ) − v ( t 0 ) divided by the time δt = t 1 − t 0 taken to make this change: a = δv δt This is an exact formula because the car was accelerating uniformly over the time between t 0 and t 1 .
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
