Mathematics
Linear Differential Equation
A linear differential equation is an equation involving a function and its derivatives, where the function and its derivatives appear only in a linear manner. This means that the function and its derivatives are raised to the power of 1 and are not multiplied or divided by each other. Linear differential equations are important in various fields such as physics, engineering, and economics for modeling natural phenomena.
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eBook - PDFIntroduction to Ordinary Differential Equations
Academic Press International Edition
- Albert L. Rabenstein(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
CHAPTER 1 Linear Differential EquationS LI Introduction An ordinary differential equation is simply an equation that involves a single unknown function, of a single variable, and some finite number of its deriva-tives. Examples of differential equations for an unknown function y(x) are dy n (a) -f. + xy 2 =2x 9 ax d 2L + e * d -l. dx 2 dx ( b ) 1Z2 + * x Ì7.-y = °· The order of a differential equation is the order of the highest order derivative of the unknown function that appears in the equation. The orders of the equations in the above examples are one and two, respectively. The adjective ordinary is used to distinguish a differential equation from one that involves an unknown function of several variables, along with the partial derivatives of the function. Equations of this latter type are called partial differential equations. An example of a partial differential equation for a function u(x,t) of two variables is d 2 u d 2 u du BF = Ô? + 2 Ô-X + U -Except for Chapter 11, this book concerns itself mainly with ordinary differen-tial equations. A linear ordinary differential equation is an equation of the special form d n y d n ~ l y dy floW j^n + *ι( χ ) -J^ï + ··' +a n -l (x)— + a n (x)y =/(*), (1.1) 3 4 I Linear Differential Equations where the functions a^x) and f(x) are given functions. The functions a^x) are called the coefficients of the equation. When f(x) = 0, the equation is said to be homogeneous; otherwise it is said to be nonhomogeneous. It is with equations of the form (1.1) that we shall be mainly concerned in this chapter.
eBook - PDFMathematics N6 Student's Book
TVET FIRST
- SA Chuturgoon JV John(Author)
- 2018(Publication Date)
- Troupant(Publisher)
differential equation: an equation that contains the derivative or derivatives of a function (and very often the functions themselves too) Differential equations TVET FIRST 113 CHALLENGE/ENRICHMENT In a Linear Differential Equation: • The power of any derivative must be 1. • The power of the dependent variable must be 1. • There is no product of the dependent variable and its derivative. The following are not linear equations: • ( dy __ dx ) 2 + 4 y = 3 x [The degree of the derivative is 2.] • d 2 y ___ dx 2 + 4 y 2 = 3 x [The power of the dependent variable, y , is 2, so more than 1.] • y dy __ dx + sin x = tan x [There is a product of the dependent variable, y , and its derivative.] The following are linear equations: • dy __ dx + 4 y = 3 x [Linear and of the first order, involving only the first derivative, dy __ dx .] • x dy __ dx + y = e x Linear and of the first order, where x dy __ dx is the product of the independent variable, x , and the derivative of the dependent variable, y . • d 2 y ___ dx 2 + dy __ dx + 2 y = 3 x Linear and of the second order, involving the unknown function, y , its derivatives and the variable x . We know that solving an equation means to find an unknown variable in the equation. Solving a differential equation has a similar but slightly different meaning. Solving a differential equation is to find the unknown function of which the derivative(s) appear in the equation. However, it is not always possible to find the function itself. If we can indeed find the function, we call the solution an explicit solution. Sometimes we can only find a relation between the dependent variable and the independent variable by getting rid of the derivatives. When this is the case, we call the solution an implicit solution. Finding a function from its derivative(s) involves integration. Therefore, solving differential equations is an application of integration. A derivative is the rate at which a certain quantity changes with respect to another.
- 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.
eBook - PDF- Joseph M. Powers, Mihir Sen(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
4 Linear Ordinary Differential Equations We consider in this chapter linear ordinary differential equations. We have already introduced first-order Linear Differential Equations in Chapter 3. Here we are mainly concerned with equations that are second order or higher in a single dependent variable. We review several topics that are commonly covered in undergraduate mathematics, including complementary functions, particular solutions, the superpo- sition principle, Sturm-Liouville equations, and resonance of a sinusoidally forced linear oscillator. We close with a discussion of linear difference equations. Strictly speaking, these are not differential equations, but they certainly arise in many dis- cretized forms of Linear Differential Equations, and their solution has analog to the solution of differential equations. Intrinsic in much of our discussion will be the notion of oscillation at a variety of frequencies. This lays the foundation of the ex- ercise of seeking repetitive patterns, a topic of relevance in engineering. The chapter also introduces the important concept of representation of a function by infinite trigonometric and nontrigonometric Fourier series, as well as projection of a func- tion onto a basis composed of a finite Fourier series. This motivates important abstractions that will be considered in detail in the later Chapter 6. Advanced topics such as Green’s functions for particular solutions and discrete/continuous spectra are included as well. All of these topics have relevance in the wide assortment of engineering systems that are well modeled by linear systems. We will provide some focus on linear oscillators, such as found in mass-spring systems. Analogs abound and are too numerous to be delineated. 4.1 Linearity and Linear Independence An ordinary differential equation can be written in the form Ly = f (x), (4.1) where L is a known operator, y is an unknown function, and f (x) is a known function of the independent variable x.
- 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.
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- Wei-Chau Xie(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
4 C H A P T E R Linear Differential Equations 4.1 General Linear Ordinary Differential Equations In general, an n th-order linear ordinary differential equation is of the form a n ( x ) d n y d x n + a n − 1 ( x ) d n − 1 y d x n − 1 + · · · + a 1 ( x ) d y d x + a 0 ( x ) y = F ( x ) , (1) in which the dependent variable y and its derivatives of various orders y , d y d x , d 2 y d x 2 , . . . , d n y d x n appear linearly in the differential equation. The coefficients a 0 ( x ) , a 1 ( x ) , . . . , a n ( x ) and the right-hand side F ( x ) are functions of x only. If the coefficients a 0 , a 1 , . . . , a n are constants, then equation (1) is a linear ordinary differential equation with constant coefficients. The D -Operator The D -operator is defined as D y ≡ d y d x , D y is taking first-order derivative of y w.r.t. x . D 2 y = D ( D y ) = d 2 y d x 2 , D 2 y is taking second-order derivative of y w.r.t. x . · · · · · · D n y = d n y d x n , n is a positive integer. D n y is taking n th-order derivative of y w.r.t. x . 140 4.1 general linear ordinary differential equations 141 Hence the D -operator is a differential operator; applying the D -operator on func-tion f ( x ) means differentiating f ( x ) with respect to x , i.e., D f ( x ) = d f ( x ) d x . Properties of the D -Operator The following properties of the D -operator can be easily verified: ( 1 ) D y 1 ( x ) + y 2 ( x ) = d d x ( y 1 + y 2 ) = d y 1 d x + d y 2 d x = D y 1 + D y 2 ; ( 2 ) D c y ( x ) = d d x ( c y ) = c d y d x = c D y , c = constant; ( 3 ) D c 1 y 1 ( x ) + c 2 y 2 ( x ) = c 1 D y 1 + c 2 D y 2 , c 1 , c 2 = constants .
- Andrei D. Polyanin, Alexander V. Manzhirov(Authors)
- 2006(Publication Date)
- Chapman and Hall/CRC(Publisher)
This follows from the fact that differential equations have in fi nitely 14.2. B ASIC P ROBLEMS OF M ATHEMATICAL P HYSICS 591 many particular solutions. The speci fi c solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions. Throughout this section, we consider linear equations in the n -dimensional Euclidean space R n or in an open domain V R n (exclusive of the boundary) with a suf fi ciently smooth boundary S = ∂V . 14.2.1-1. Parabolic equations. Initial and boundary conditions. In general, a linear second-order partial differential equation of the parabolic type with n independent variables can be written as ∂w ∂t – L x , t [ w ] = Φ ( x , t ), (14. 2 . 1 . 1 ) where L x , t [ w ] ≡ n i , j = 1 a ij ( x , t ) ∂ 2 w ∂x i ∂x j + n i = 1 b i ( x , t ) ∂w ∂x i + c ( x , t ) w , (14. 2 . 1 . 2 ) x = { x 1 , . . . , x n } , n i , j = 1 a ij ( x , t ) ξ i ξ j ≥ σ n i = 1 ξ 2 i , σ > 0 . Parabolic equations govern unsteady thermal, diffusion, and other phenomena dependent on time t . Equation (14.2.1.1) is called homogeneous if Φ ( x , t ) ≡ 0 . Cauchy problem ( t ≥ 0 , x R n ). Find a function w that satis fi es equation (14.2.1.1) for t > 0 and the initial condition w = f ( x ) at t = 0 . (14. 2 . 1 . 3 ) Boundary value problem * ( t ≥ 0 , x V ). Find a function w that satis fi es equa-tion (14.2.1.1) for t > 0 , the initial condition (14.2.1.3), and the boundary condition Γ x , t [ w ] = g ( x , t ) at x S ( t > 0 ). (14. 2 . 1 . 4 ) In general, Γ x , t is a fi rst-order linear differential operator in the space variables x with coef fi cient dependent on x and t . The basic types of boundary conditions are described in Subsection 14.2.2. The initial condition (14.2.1.3) is called homogeneous if f ( x ) ≡ 0 . The boundary condition (14.2.1.4) is called homogeneous if g ( x , t ) ≡ 0 .
- D.S. Jones, Michael Plank, B.D. Sleeman(Authors)
- 2009(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 3 Systems of Linear Ordinary Differential Equations 3.1 First-order systems of equations with constant coefficients In studying natural phenomena, we are often interested in more than one quantity and the several quantities may well be connected by differential equa-tions. We are therefore led to consider what happens when more than one differential equation has to be solved at a time. Suppose a 1 ˙ x + b 1 ˙ y + c 1 x + d 1 y = f 1 ( t ) , (3.1.1) a 2 ˙ x + b 2 ˙ y + c 2 x + d 2 y = f 2 ( t ) (3.1.2) where a 1 , b 1 , c 1 , d 1 , a 2 , b 2 , c 2 , d 2 are constants and x , y are to be found. In other words, two simultaneous differential equations of the first order have to be solved. Multiply (3.1.1) by b 2 and (3.1.2) by b 1 . Then subtraction gives ( a 1 b 2 − a 2 b 1 ) ˙ x + αx + βy = F ( t ) (3.1.3) where α = c 1 b 2 − c 2 b 1 , β = d 1 b 2 − d 2 b 1 and F ( t ) = b 2 f 1 ( t ) − b 1 f 2 ( t ). There are two distinct cases to discuss according as a 1 b 2 − a 2 b 1 is or is not zero. We call a 1 b 2 − a 2 b 1 the test determinant . 3.1.1 Test determinant is nonzero If β negationslash = 0, (3.1.3) can be solved to give y in terms of x and ˙ x . If this expression is substituted in (3.1.1) or (3.1.2) a Linear Differential Equation with constant coefficients of order 2 is obtained for x . This differential equation can be solved by techniques already described and its general solution will involve two arbitrary constants. Having found x we can determine y from (3.1.3). No further arbitrary constants are introduced and so the whole solution contains two arbitrary constants. If β = 0, (3.1.3) is a differential equation of the first order for x which can be resolved by means of an integrating factor. Its general solution will possess 61 62 Differential Equations and Mathematical Biology one arbitrary constant. Once x is known, it can be substituted in (3.1.1) or (3.1.2) resulting in a differential equation of the first order for y .
eBook - PDF- David Pearson(Author)
- 1995(Publication Date)
- Butterworth-Heinemann(Publisher)
In Section 5.1 we saw that differentiation may be thought of as a linear operator, in the sense that the derivative of Af, + Bf2 is given by ~ (Afl + Bh) = A dfl + B dh dx dx dx for any constants A and B. The linearity of differentiation is important in many aspects of the calculus of functions, but nowhere more so than in the theory of DEs; and especially in the theory of Linear Differential Equations. What is a linear DE? To take first of all the simplest case, a linear DE of first order is an equation df/dx == Q(x,f) where for each value of x the right-hand side function depends linearly on f That is, Q == af + b, so the DE takes the form df/dx == af + b. Here a and b are constants as far asfis concerned, that is, they do not depend explicitly on/; they may, however, depend on x, so we should write a == a(x), b == b(x), and the DE becomes dj/dx == a(x)f + b(x). (Of course a or b or both may be constant functions, or even the zero function, but these are special cases.) A linear DE of second order is an equation d 2f/dx 2 == Q(x,/, df/dx) where for each value of x the right-hand side function depends linearly on bothfand df/dx. The DE then looks like d 2f/dx 2 == a(x)df/dx + b(x)f + c(x), where a, band care given functions of x. Linear DEs of order greater than 2 are defined by an obvious extension of the same idea. A common convention is to take all terms involving f or derivatives off to the left-hand side of the equation. DEs of first and second order are then written, respectively, as :~ -a(x)f = b(x) and d 2 f df dx 2 -a(x) dx -b(x)f == c(x) A more systematic notation is needed, particularly if we are to deal with higher order equations. We shall generally denote b~ Po (x) the coefficient off, PI (x) tpe coefficient of *f, P2(X) the coefficient of ~, and Pk(X) the coefficient of ~J; the right-hand side function will be denoted by p(x). Then the general first order linear DE becomes PI (x) :~ + Po (x)f = p(x)
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