Mathematics
Reducible Differential Equations
Reducible differential equations are equations that can be simplified or transformed into a more manageable form through a change of variables or other techniques. By reducing the complexity of the equation, it becomes easier to solve and analyze. This process often involves identifying specific substitutions or transformations that lead to a simpler form of the original equation.
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3 Key excerpts on "Reducible Differential Equations"
- 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)
- 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.
- 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.
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