Mathematics
General Solution of Differential Equation
The general solution of a differential equation is a family of solutions that includes all possible specific solutions to the equation. It typically contains arbitrary constants that can take on different values, allowing for a wide range of specific solutions. Finding the general solution is important in solving differential equations and understanding the behavior of the system they model.
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7 Key excerpts on "General Solution of Differential Equation"
eBook - PDF- Maria Catherine Borres(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
c. Study of chemical reactions. d. Determination of curves with given geometrical properties. 8.6. SOLUTION OF A DIFFERENTIAL EQUATION A solution (or integral) of a differential equation is a relation between the variables, not containing derivatives, such that this relation and the derivatives obtained from it satisfy the given differential equation identically. For example, the equation =− dy y dx µ has a solution . , − = x y c e µ where c is an arbitrary constant. The differential equation 2 2 0 + = d y y dx has solution cos , sin = = y A x y B x and cos sin , = + y A x B x where Aand B are arbitrary constants. A solution of a differential equation which contains as many arbitrary con -stants as the order of the equation is called a general solution (or integral) of the differential equation. A solution obtained from the general solution by giving particular values to the constants is called a particular solution. The graph of a particular integral is called an integral curve of the differential equation. 8.7. FORMATION OF A DIFFERENTIAL EQUA-TION Given a relation ( ) 1 2 , , , , , 0 … = n f x y c c c (1) between variables , x y and containing n constants 1 2 , , , … n c c c it is always possible to form a differential equation of order n such that the given rela - Principles of Applied Mathematics 300 tion (1) is the general solution of the equation. This is done by differentiating (1) n times thereby obtaining equations and then eliminating the n constants from the original relation and n derived equations. The method is illustrated by means of examples. Example: The equation = + y x a (1) Represents a family of parallel straight lines for different values of . a Elimination of one constant ‘a’ Requires two equations. The second equation is obtained by differentiating (1). Thus 1 = dy dx is the differential equation of the relation (1), with a eliminated.
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)
In this subsection, our goal is to discuss a procedure for finding a general solution to (2.23). This provides a basis for analyzing, understanding, and solving problems in the biological, chemical, physical, and social sciences. Definition 2.4.1. Let J = [ a, b ] for a, b ∈ R and hence J ⊆ R be an interval. A solution process for the first-order linear scalar deterministic differential equation (2.23) is a function x defined on J into R such that it satisfies (2.23) on J in the sense of elementary calculus. In short, if we substitute a function x ( t ) and its differential dx ( t ) into (2.23), then the equation remains valid for all t in J . Example 2.4.1. Verify that x ( t ) = 5 e at is a solution to the differential equation dx = ax dt , where a is any given real number. Moreover, for any arbitrary constant c , show that x ( t ) = ce at is also a solution to the given differential equation. Solution procedure. We apply deterministic differential calculus to x ( t ) = 5 e at , and we compute the following differential: dx ( t ) = d ( 5 e at ) (from the given expression) = 5 ae at dt (by properties of the differential and the chain rule) = ax dt (by substitution for 5 e at ) . 110 An Introduction to Differential Equations: Vol. 1 This shows that the given function x ( t ) = 5 e at satisfies the given differential equa-tion. Furthermore, by repeating the above argument, one can show that x ( t ) = ce at , and it also satisfies the same differential equation. We note that the “5” is replaced by an arbitrary constant c . From Definition 2.4.1, we conclude that the given func-tions are solutions to the given differential equation. 2.4.2 Procedure for finding a general solution We introduce an eigenvalue–eigenvector-like approach to solving linear homogeneous scalar differential equations. Of course, this may sound complicated. However, the underlying ideas are very simple, and it is motivated by the knowledge of solving lin-ear algebraic equations.
eBook - PDFDifferential Equations
An Introduction to Basic Concepts, Results and Applications
- Ioan I Vrabie(Author)
- 2004(Publication Date)
- WSPC(Publisher)
system. It is easy to realize that, in order to get a fairly acceptable prediction close enough to the reality, we need fairly precise data on the present state of the system, as well as, sound knowledge on the law(s) according to which the instantaneous state of the system affects its instantaneous rate of change. Mathematical Modelling is that discipline which comes into play at this point, offering the scientist the description of such laws in a mathematical language, laws which, in many specific situations, take the form of differential equations, or even of systems of differential equations. The goal of the present section is to define the concept of differential equation, as well as that of system of differential equations, and to give a brief review of the main problems to be studied in this book. Roughly speaking, a scalar diflerential equation represents a functional dependence relationship between the values of a real valued function, called unknown function, some, but at least one of its ordinary (partial) derivatives up to a given order n, and the independent variable(s). The highest order of differentiation of the unknown function involved in the equation is called the order of the equation. A differential equation whose unknown function depends on one real variable is called ordinary diflerential equation, while a differential equa- tion whose unknown function depends on two, or more, real independent variables is called a partial diflerential equation. For instance the equation x“ + x = sin t, whose unknown function x depends on one real variable t, is an ordinary differential equation of second order, while the equation 12 Generalities whose unknown function u depends on two independent real variables x and y, is a third-order partial differential equation.
eBook - PDFLearning and Teaching Mathematics using Simulations
Plus 2000 Examples from Physics
- Dieter Röss(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
They also have a wide range of applicability as well as high aesthetic appeal, and provide order in the plethora of natural phenomena. 9.3 Solution methods for ordinary differential equations If one wants to obtain from the differential equation a closed form for y.t/ , then it needs to be solved analytically , as shown for the particularly easy case of the expo-nential function. If the function y does not appear on the right-hand side of the dif-ferential equation, i.e. if the differential equation reads y 0 D f .t/ , then y is obtained via the normal integration method. We can immediately verify this for the traveled distance problem. For constant acceleration b we obtain for the traveled distance s.t/ the following: s 00 D b I s 0 D v D Z t 0 bdt 0 D b Z t 0 dt 0 D bt C v 0 I s D Z t 0 .bt 0 C v 0 /dt 0 D bt 2 2 C v 0 t C s 0 : Here the two initial values are the initial position s 0 and the initial velocity v 0 . 9.4 Numerical solution methods: initial value problem 170 For the general case, the art of solving differential equations analytically fills entire books. The solution methods for those differential equations that are important in physics mostly follow simple patterns, for which there are standard methods. We refer here to a few of the books cited in the introduction. In general, all ordinary differential equations can be treated analytically. For this endeavor, an approach quite similar to the integration of non-standard functions is applied: one tries to guess a specific solution systematically and then tries to obtain a general solution with the variation or determination of parameters, which can be an exact, an approximate or a series solution of the differential equation.
eBook - PDFComputational Mathematics in Engineering and Applied Science
ODEs, DAEs, and PDEs
- W.E. Schiesser(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
chapter one The General Problems in Ordinary, Differential Algebraic and Partial Differential Equations Ordin~~ry differential equations (ODEs), differential algebraic equations (DAEs), and partial differential equations (PDEs) are mathematical forms that have broad application in sci-ence and engineering. The formulation of mathematical models expressed as sets of ODEs/DAEs/PDEs is frequently the starting point for quantatitive studies of the be-havior and performance of scientific and engineering systems. Mathematical models, however, have limited utility unless solutions to the models can be produced and studied with reasonable effort. Before the general purpose scientific computer became available, models were typically manipulated analytically, often with major assump-tions made during the analysis so that the mathematics would be tractable. Practically, this meant that models consisting of at most a few ODEs/DAEs/PDEs could be solved, and this process often required great ingenuity, particularly in the case of nonlinear equations. With the availability of the scientific computer, this general requirement for mod-els to be highly simplified was completely circumvented; now large sets of nonlinear ODEs/DAEs/PDEs, in principle, can be integrated numerically. In practice, the cod-ing (programming) of a numerical algorithm to solve a particular ODE/DAE/PDE problem system can appear daunting to the scientist or engineer who has limited knowledge and experience in numerical mathematics, and who wishes primarily to arrive at a useful solution without a major investment of time and effort to learn the details of numerical analysis and programming. Additionally, even if the analyst is willing to make this investment of time and effort, the direction this effort should take is often far from clear.
- 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 .
- K. F. Riley, M. P. Hobson(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
T. H. Piaggio, An Elementary Treatise on Differential Equations and their Applications (London: G. Bell and Sons, Ltd, 1954), pp. 175 ff. 399 10.3 General and particular solutions integral of ( 10.18 ) is u ( x, y ) = − 3 y , and so the general solution to ( 10.18 ) is u ( x, y ) = f ( x 2 + y 2 ) − 3 y. Boundary condition (i) requires u ( x, 0) = f ( x 2 ) = x 2 , i.e. f ( z ) = z , and so the particular solu-tion in this case is u ( x, y ) = x 2 + y 2 − 3 y. Similarly, boundary condition (ii) requires u (1 , 0) = f (1) = 2. One possibility is f ( z ) = 2 z , and if we make this choice, then one way of writing the most general particular solution is u ( x, y ) = 2 x 2 + 2 y 2 − 3 y + g ( x 2 + y 2 ) , where g is any arbitrary function for which g (1) = 0. Alternatively, a simpler choice would be f ( z ) = 2, leading to u ( x, y ) = 2 − 3 y + h ( x 2 + y 2 ) , where, this time, h (1) = 0. Clearly, if the two solutions are to represent the same explicit solution, we must have that h ( z ) = g ( z ) + 2( z − 1), but, for the most general solution satisfying this one-point boundary condition, either form will do. Although we have discussed the solution of inhomogeneous problems only for first-order equations, the general considerations hold true for linear PDEs of higher order. 10.3.3 Second-order equations As noted in Section 10.1 , second-order linear PDEs are of great importance in describing the behavior of many physical systems. As in our discussion of first-order equations, for the moment we will restrict our discussion to equations with just two independent variables; extensions to a greater number of independent variables are straightforward. The most general second-order linear PDE (containing two independent variables) has the form A ∂ 2 u ∂x 2 + B ∂ 2 u ∂x∂y + C ∂ 2 u ∂y 2 + D ∂u ∂x + E ∂u ∂y + Fu = R ( x, y ) , (10.19) where A, B, . . . , F and R ( x, y ) are given functions of x and y .
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