Mathematics
Coupled First-order Differential Equations
Coupled first-order differential equations refer to a system of two or more first-order differential equations that are interconnected and influence each other's behavior. These equations are typically solved simultaneously to find a solution that satisfies all of them. Coupled systems often arise in various fields of science and engineering, and their solutions can provide valuable insights into complex dynamic systems.
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9 Key excerpts on "Coupled First-order Differential Equations"
- Stephen A. Wirkus, Randall J. Swift, Ryan Szypowski(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
They can often accurately capture the behavior of continuous models or a large number of discrete objects where the current state of the system determines the future behavior of the system. Such models are called deterministic (as opposed to stochastic or random). The study of nonlinear differential equations is still a very active area of research. Although this text will consider some nonlinear differential equations, here the focus will be on the linear case. We will begin with some basic terminology. 1.1 Introduction to First-Order Equations Order, Linear, Nonlinear We begin our study of differential equations by explaining what a differential equation is. From our experience in calculus, we are familiar with some differential equations. For example, suppose that the acceleration of a falling object is a(t) = -32, measured in ft/sec 2 . Using the fact that the derivative of the velocity function v(t) (measured in ft/sec) is the acceleration function a(t), we can solve the equation dv v (t) = a(t) or = a(t) = -32. dt Many different types of differential equations can arise in the study of familiar phenomena in subjects ranging from physics to biology to economics to chemistry. We give examples from various fields throughout the text and engage the reader with many such applications. 1 2 Chapter 1. Traditional First-Order Differential Equations It is clearly necessary (and expedient) to study, independently, more restricted classes of these equations. The most obvious classification is based on the nature of the derivative(s) in the equation. A differential equation involving derivatives of a function of one variable (ordinary derivatives) is called an ordinary differential equation, whereas one containing partial derivatives of a function of more than one independent variable is called a partial differential equation. In this text, we will focus on ordinary differential equations.
eBook - PDFOrdinary Differential Equations
A Practical Guide
- Bernd J. Schroers(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
1 First order differential equations 1.1 General remarks about differential equations 1.1.1 Terminology A differential equation is an equation involving a function and its derivatives. An example which we will study in detail in this book is the pendulum equation d 2 x dt 2 = - sin(x), (1.1) which is a differential equation for a real-valued function x of one real variable t. The equation expresses the equality of two functions. To make this clear we could write (1.1) more explicitly as d 2 x dt 2 (t) = - sin(x(t)) for all t ∈ R, but this would lead to very messy expressions in more complicated equations. We therefore often suppress the independent variable. When formulating a mathematical problem involving an equa- tion, we need to specify where we are supposed to look for solu- tions. For example, when looking for a solution of the equation x 2 +1 = 0 we might require that the unknown x is a real number, in which case there is no solution, or we might allow x to be com- plex, in which case we have the two solutions x = ±i. In trying to solve the differential equation (1.1) we are looking for a twice- differentiable function x : R → R. The set of all such functions is very big (bigger than the set of all real numbers, in a sense that can be made precise by using the notion of cardinality), and this 1 2 First order differential equations is one basic reason why, generally speaking, finding solutions of differential equations is not easy. Differential equations come in various forms, which can be clas- sified as follows. If only derivatives of the unknown function with respect to one variable appear, we call the equation an ordinary differential equation, or ODE for short. If the function depends on several variables, and if partial derivatives with respect to at least two variables appear in the equation, we call it a partial differen- tial equation, or PDE for short.
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 - PDF- Stephen A. Wirkus, Randall J. Swift(Authors)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
They can often accurately capture the behavior of continuous models or a large number of discrete objects where the current state of the system determines the future behavior of the system. Such models are called deterministic (as opposed to stochastic or random ). The study of nonlinear differential equations is still a very active area of research. Although this text will consider some nonlinear differential equations, here the focus will be on the linear case. We will begin with some basic terminology. 1.1 Introduction to First-Order Equations Order, Linear, Nonlinear We begin our study of differential equations by explaining what a differen-1 2 Chapter 1. Traditional First-Order Differential Equations tial equation is. From our experience in calculus, we are familiar with some differential equations. For example, suppose that the acceleration of a falling object is a ( t ) = -32, measured in ft/sec 2 . Using the fact that the derivative of the velocity function v ( t ) (measured in ft/sec) is the acceleration function a ( t ), we can solve the equation v 0 ( t ) = a ( t ) or dv dt = a ( t ) = -32 . Many different types of differential equations can arise in the study of familiar phenomena in subjects ranging from physics to biology to economics to chem-istry. We give examples from various fields throughout the text and engage the reader with many such applications. It is clearly necessary (and expedient) to study, independently, more re-stricted classes of these equations. The most obvious classification is based on the nature of the derivative(s) in the equation. A differential equation in-volving derivatives of a function of one variable (ordinary derivatives) is called an ordinary differential equation , whereas one containing partial deriva-tives of a function of more than one independent variable is called a partial differential equation . In this text, we will focus on ordinary differential equations.
- 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.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Each of these statements involves a rate of 40 CHAPTER 2 First-Order Differential Equations change (derivative) and consequently, when expressed mathematically, leads to a differential equation. The differential equation is a mathematical model of the process. It is important to realize that the mathematical equations are almost always only an approximate description of the actual process. For example, bodies moving at speeds comparable to the speed of light are not governed by Newton’s laws, insect populations do not grow indefinitely as stated because of eventual lack of food or space, and heat transfer is affected by factors other than the temperature difference. Thus you should always be aware of the limitations of the model so that you will use it only when it is reasonable to believe that it is accurate. Alternatively, you can adopt the point of view that the mathematical equations exactly describe the operation of a simplified physical model, which has been constructed (or conceived of) so as to embody the most important features of the actual process. Sometimes, the process of mathematical modeling involves the conceptual replacement of a discrete process by a continuous one. For instance, the number of members in an insect population changes by discrete amounts; however, if the population is large, it seems reasonable to consider it as a continuous variable and even to speak of its derivative. Step 2: Analysis of the Model. Once the problem has been formulated mathematically, you are often faced with the problem of solving one or more differential equations or, failing that, of finding out as much as possible about the properties of the solution. It may happen that this mathematical problem is quite difficult, and if so, further approximations may be indicated at this stage to make the problem mathematically tractable. For example, a nonlinear equation may be approximated by a linear one, or a slowly varying coefficient may be replaced by a constant.
- Yuefan Deng(Author)
- 2017(Publication Date)
- World Scientific(Publisher)
1 Chapter 1 First‐Order Differential Equations 1.1 Definition of Differential Equations A differential equation ሺDEሻ is a mathematical equation that relates some functions of one or more variables with their derivatives. A DE is used to describe changing quantities and it plays a major role in quantitative studies in many disciplines such as all fields of engineering, physical sciences, life sciences, and economics. Examples Are they DEs or not? ܽ ݔ ଶ ܾ ݔ ܿ ൌ 0 No Chapter 1 First‐Order Differential Equations 2 ܽ ݔ ଶ ܾ ݔ ′ ܿ ൌ 0 Yes Here ݔ ′ ൌ ௗ ௫ ௗ ௧ ܽ ݔ ଶ ܾ ݔ ′ ܿ ݕ ′ ൌ 0 Yes Here ݔ ′ ൌ ௗ ௫ ௗ ௧ and ݕ ′ ൌ ௗ ௬ ௗ ௧ ݕ ′′ ൌ ݔ ଷ Yes Here ݕ ′ ൌ ௗ ௬ ௗ ௧ To solve a DE is to express the solution of the unknown function ሺthe dependent variable or DVሻ in mathematical terms without the derivatives. Examples ܽ ݔ ᇱ ܾ ൌ 0 ݔ ′ ൌ െ ܾ ܽ is not a solution ݔ ൌ െ න ܾ ܽ ݀ ݐ is a solution In general, there are two common ways in solving DEs: analytically and numerically. Most DEs, difficult to solve by analytical methods, must be “solved” by using numerical methods, although many DEs are too stiff to solve by 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 to make the contents of the book flow but one may still need to look up terms undefined here or abbreviations introduced at the end of the book. First, we introduce the terms of dependent variables ሺDVsሻ and independent 1.1 Definition of Differential Equations 3 variables ሺIVsሻ of functions.
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.
eBook - PDFDifferential Equations
An Introduction to Modern Methods and Applications
- James R. Brannan, William E. Boyce(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
These features can be interpreted in terms of the physical behavior of the systems that the differential equations model. Furthermore, it is often easy to vary parameters in the mathe- matical model over wide ranges, whereas this may be very time-consuming or expensive, if 56 Chapter 2 First Order Differential Equations not impossible, in an experimental setting. Nevertheless mathematical modeling and exper- iment or observation are both critically important and have somewhat complementary roles in scientific investigations. Mathematical models are validated by comparison of their pre- dictions with experimental results. On the other hand, mathematical analyses may suggest the most promising directions for experimental exploration, and they may indicate fairly precisely what experimental data will be most helpful. In Section 1.1 we formulated and investigated a few simple mathematical models. We begin by recapitulating and expanding on some of the conclusions reached in that section. Regardless of the specific field of application, there are three identifiable stages that are always present in the process of mathematical modeling. ▶ Construction of the Model. In this stage, you translate the physical situation into mathematical terms, often using the steps listed at the end of Section 1.1. Perhaps most critical at this stage is to state clearly the physical principle(s) that are believed to govern the process. For ex- ample, it has been observed that in some circumstances heat passes from a warmer to a cooler body at a rate proportional to the temperature difference, that objects move about in accordance with Newton’s laws of motion, and that isolated insect populations grow at a rate proportional to the current population. Each of these statements involves a rate of change (derivative) and consequently, when expressed mathematically, leads to a differen- tial equation. The differential equation is a mathematical model of the process.
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