Mathematics
Second-order Differential Equations
Second-order differential equations are mathematical equations that involve the second derivative of an unknown function. They are commonly used to model physical systems and phenomena in fields such as physics and engineering. Solutions to second-order differential equations often involve finding the general solution and applying initial conditions or boundary conditions to obtain specific solutions.
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10 Key excerpts on "Second-order 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.
eBook - PDF- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2016(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- Virginia W. Noonburg(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
8 Solving Second-order Partial Differential Equations This chapter focuses on second-order partial differential equations in two variables. These include the classical equations of physics: the heat equation, wave equation, and Laplace’s equation. It will be seen that these three equations are canonical examples of what are referred to as parabolic, hyperbolic, and elliptic PDEs. All three of these equations are linear and can be solved by the method of separation of variables. Differ-ent initial values and boundary conditions will be shown to require slightly different techniques. To solve problems encountered in the biological sciences it is often necessary to work with nonlinear PDEs and, for these, numerical methods are often required. A numerical method will be given for each of the three types of linear PDEs, and it will be shown that these also apply to simple nonlinear PDEs. In the final section of the chapter a student project involving a PDE appearing in the mathematical biology literature is treated in detail. 8.1 Classification of Linear Second-order Partial Differential Equations Most of the partial differential equations we will consider will be linear equations of second order; where the unknown function ?(?, ?) is a function of two independent variables ? and ? , or sometimes ? and ? , when the variable ? is used to represent time. The most general second-order linear partial differential equation in two inde-pendent variables ? and ? can be written in the form ?? ?? + 2?? ?? + ?? ?? + ?? ? + ?? ? + ?? = ? (8.1) where ?, ?, … , ? can be arbitrary functions of ? and ? . The coefficient of ? ?? is written as 2? since we are assuming that for the functions ? that we will be dealing with, the mixed partial derivatives ? ?? and ? ?? will be equal. Any equation that can be put in 285 286 Chapter 8 Solving Second-order Partial Differential Equations this form is called linear , and if ?(?, ?) ≡ 0 it is called a homogeneous linear PDE .
eBook - PDF- George Turrell(Author)
- 2001(Publication Date)
- Academic Press(Publisher)
5 Ordinary Differential Equations Differential equations are usually classified as “ordinary” or “partial”. In the former case only one independent variable is involved and its differential is exact. Thus there is a relation between the dependent variable, say y ( x ), its various derivatives, as well as functions of the independent variable x . Partial differential equations contain several independent variables, and hence partial derivatives. The order of an ordinary differential equation is the order of its highest derivative. Thus, an ordinary differential equation of order n is an equation of the form F(x, y, y , . . . , y (n) ) = 0 . ( 1 ) If the dependent variable y ( x ) and all of its derivatives occur in the first degree and do not appear as products, the equation is said to be linear. In effect, the solution of a differential equation of order n necessitates n integrations, each of which involves an arbitrary constant. However, in some cases one or more of these constants may be assigned specific values. The results, which are also solutions of the differential equation, are referred to as particular solutions. The general solution, however, includes all of the n constants of integration, whose evaluation requires additional information associated with the application. 5.1 FIRST-ORDER DIFFERENTIAL EQUATIONS A first-order differential equation can always be solved, although its solution is not necessarily easy to obtain. If the variables are separable, the equation can be reduced to the form f (x) d x = g(y) d y, ( 2 ) and the integration can usually be carried out by one of the methods illustrated in Section 3.3. Furthermore, as shown in Section 3.5, a differential equation such as N( x,y ) d x + M( x,y ) d y = 0 ( 3 ) 86 MATHEMATICS FOR CHEMISTRY AND PHYSICS can be integrated directly if the left-hand side is an exact differential.
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)
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. It is important to realize that the mathematical equations are almost always only an ap- proximate description of the actual process. For example, bodies moving at speeds compa- rable to the speed of light are not governed by Newton’s laws, insect populations do not grow indefinitely as stated because of eventual limitations on their food supply, and heat transfer is affected by factors other than the temperature difference. Alternatively, one can adopt the point of view that the mathematical equations exactly describe the operation of a simpli- fied or ideal physical model, which has been constructed (or imagined) 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 is an integer; however, if the population is large, it may seem reasonable to consider it to be a continuous variable and even to speak of its derivative. ▶ 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 mathe- matical problem is quite difficult, and if so, further approximations may be required at this stage to make the problem more susceptible to mathematical investigation. For example, a nonlinear equation may be approximated by a linear one, or a slowly varying coefficient may be replaced by a constant.
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.
eBook - PDF- Steven Holzner(Author)
- 2008(Publication Date)
- For Dummies(Publisher)
143 Chapter 6: Studying Second Order Linear Nonhomogeneous Differential Equations In fact, it turns out that the denominator is the Wronskian (introduced in Chapter 5), W , for y 1 , y 2 , and x , W ( y 1 , y 2 )( x ): W = y 1 ( x ) y' 2 ( x ) – y 1 ' ( x ) y 2 ( x ) So, you can write the equations for u' 1 ( x ) and u' 2 ( x ) like this instead: u x y x g x W y y x 1 2 1 2 , and: u x y x g x W y y x 2 1 1 2 , Note that dividing by the Wronskian is okay because y 1 and y 2 are a set of lin-early independent solutions, so their Wronskian is nonzero. This means that you can solve for u 1 ( x ) (at least theoretically) like this: u x y x g x W y y x dx c 1 2 1 2 1 , And you can solve for u 2 ( x ) like this: u x y x g x W y y x dx c 2 1 1 2 2 , Of course, there’s no guarantee that you can perform the integrals in these equations. But if you can, you can get u 1 ( x ) and u 2 ( x ), such that a particular solution to the differential equation is: y p = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) The general solution is: y = c 1 y 1 ( x ) + c 2 y 2 ( x ) + u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) Bouncing Around with Springs ’n’ Things Second order differential equations play a big part in elementary physics. They’re used in describing the motion of springs and pendulums, electromag-netic waves, heat conduction, electric circuits that contain capacitors and inductors, and so on. I provide a couple of examples of second order differen-tial equations in the following sections. 144 Part II: Surveying Second and Higher Order Differential Equations A mass without friction Here I show you the differential equation describing the motion of a mass on the end of a spring. Say, for example, that you have the situation shown in Figure 6-1, where a mass is moving around (without friction) on the end of a spring. In the following sections, I show you how to solve this physics prob-lem with the help of a nonhomogeneous equation. Figure 6-1: A spring with a mass moving without friction.
- Willi-hans Steeb(Author)
- 1993(Publication Date)
- World Scientific(Publisher)
Chapter 2 Second-order ordinary differential equations 2.1 Second-order ordinary differential equation Second-order ordinary differential equations and invertible point transformations have been investigated by several authors (Duarte et al 1987, Duarte et al 1989, Duarte et al 1990, Duarte et al 1991, Euler et al 1991). The starting point is a linear second order differential equation or a second order differential equation where the exact solution is known. By applying an invertible point transformation a class of nonlinear ordinary differential equations is generated. We discuss the following cases: The free-particle equation d 2 U dT 2= the harmonic oscillator S-the anharmonic oscillator g+-the anharmonic system d 2 U dT 2 + [/■ = 0, n integer and the second Painleve transcendent tPll ^L = 2U 3 + TU + a (1) (2) (3) (4) (5) where o is a constant. 7 8 CHAPTER 2. SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS The most studied case is the free particle equation Obviously, the general solution of (6) is given by U(T) = CxT + C 2 (7) where d and C 2 are the constants of integration. The invertible point transformation is given by U(T(t)) = F(u(t), t), T(t) = G(u{t), t) (8) where F and G are smooth functions (C°°) of u and t. All considerations are local. We have to assume that A : = ?rSr~£^ 0 (9) and dGdu dG . „ -du-Tt+lH**-(10) Applying the chain rule of differentiation we find that dT dGdu dG , „ , lt = -du -t t + -m ( lla) and d t/ d t; d T dU ( dGdu dG dFdu dF ,,,_, ~ d J = dT ~dl-lr 'd^li + ^)-^ u 'T t + ^-{nh) From (lib) we arrive at dFdu 9 F dr/ a -J I + ~ o7 <7U a t ot / t i .
eBook - PDF- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The method of reduction of order (Section 3.4) can also be used for the nonhomogeneous equation 𝑦 ′′ + 𝑝(𝑡)𝑦 ′ + 𝑞(𝑡)𝑦 = 𝑔(𝑡), (38) provided one solution 𝑦 1 of the corresponding homogeneous equation is known. Let 𝑦 = 𝑣(𝑡)𝑦 1 (𝑡) and show that 𝑦 satisfies equation (38) if 𝑣 is a solution of 𝑦 1 (𝑡)𝑣 ′′ + ( 2𝑦 ′ 1 (𝑡) + 𝑝(𝑡)𝑦 1 (𝑡) ) 𝑣 ′ = 𝑔(𝑡). (39) Equation (39) is a first-order linear differential equation for 𝑣 ′ . By solving equation (39) for 𝑣 ′ , integrating the result to find 𝑣, and then multiplying by 𝑦 1 (𝑡), you can find the general solution of equation (38). This method simultaneously finds both the second homogeneous solution and a particular solution. In each of Problems 24 through 26, use the method outlined in Problem 23 to solve the given differential equation. 24. 𝑡 2 𝑦 ′′ − 2𝑡𝑦 ′ + 2𝑦 = 4𝑡 2 , 𝑡 > 0; 𝑦 1 (𝑡) = 𝑡 25. 𝑡 2 𝑦 ′′ + 7𝑡𝑦 ′ + 5𝑦 = 𝑡, 𝑡 > 0; 𝑦 1 (𝑡) = 𝑡 −1 26. 𝑡𝑦 ′′ − (1 + 𝑡)𝑦 ′ + 𝑦 = 𝑡 2 𝑒 2𝑡 , 𝑡 > 0; 𝑦 1 (𝑡) = 1 + 𝑡 (see Problem 12) 3.7 Mechanical and Electrical Vibrations One of the reasons why second-order linear differential equations with constant coefficients are worth studying is that they serve as mathematical models of many important physical processes. Two important areas of application are the fields of mechanical and electrical oscillations. For example, the motion of a mass on a vibrating spring, the torsional oscilla- tions of a shaft with a flywheel, the flow of electric current in a simple series circuit, and many other physical problems are all described by the solution of an initial value problem of the form 𝑎𝑦 ′′ + 𝑏𝑦 ′ + 𝑐𝑦 = 𝑔(𝑡), 𝑦(0) = 𝑦 0 , 𝑦 ′ (0) = 𝑦 ′ 0 . (1) This illustrates a fundamental relationship between mathematics and physics: many phys- ical problems may have mathematically equivalent models.
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