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Inexact Differential Equation
An inexact differential equation is a differential equation of the form: : M(x,y) \, dx+N(x,y) \, dy=0 satisfying the condition : \frac \ne \frac Leonhard Euler invented the integrating factor in 1739 to solve these equations. Solution method To solve an inexact differential equation, it may be transformed into an exact differential equation by finding an integrating factor \mu. Multiplying the original equation by the integrating factor gives: : \mu M\,dx+\mu N\,dy=0. For this equation to be exact, \mu must satisfy the condition: : \frac=\frac. Expanding this condition gives: :M\mu_y-N\mu_x+(M_y-N_x)\mu = 0. Since this is a partial differential equation, it is generally difficult. However in some cases where \mu depends only on x or y, the problem reduces to a separable first-order linear differential equation. The solutions for such cases are: :\mu(y)=e^ or :\mu(x)=e^. See Also * Inexact differential * Exact differential equation In mathematics, an exact d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Kingdom of Prussia, Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase Pi (letter), pi) to denote Pi, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrating Factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve non-exact ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential. Use An integrating factor is any expression that a differential equation is multiplied by to facilitate integration. For example, the nonlinear second order equation : \frac = A y^ admits \frac as an integrating factor: : \frac \frac = A y^ \frac. To integrate, note that both sides of the equation may be expressed as derivatives by going backwards with the chain rule: : \frac\left(\frac 1 2 \left(\frac\right)^2\rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Differential
In multivariate calculus, a differential (infinitesimal), differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function Q in an Orthogonal coordinates, orthogonal coordinate system (hence Q is a multivariable function Dependent and independent variables#In pure mathematics, whose variables are independent, as they are always expected to be when treated in multivariable calculus). An exact differential is sometimes also called a ''total differential'', or a ''full differential'', or, in the study of differential geometry, it is termed an exact form. The integral of an exact differential over any integral path is Conservative vector field#Path independence, path-independent, and this fact is used to identify state functions in thermodynamics. Overview Definition Even if we work in three dimensions here, the definitions of exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Differential Equation
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) A differential equation for the unknown f(x) is separable if it can be written in the form :\frac f(x) = g(x)h(f(x)) where g and h are given functions. This is perhaps more transparent when written using y = f(x) as: :\frac=g(x)h(y). So now as long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, where the two variables ''x'' and ''y'' have been separated. Note ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Linear Differential Equation
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. Types of solution A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inexact Differential
An inexact differential or imperfect differential is a differential whose integral is path dependent. It is most often used in thermodynamics to express changes in path dependent quantities such as heat and work, but is defined more generally within mathematics as a type of differential form. In contrast, an integral of an exact differential is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential. I.e., its value cannot be inferred just by looking at the initial and final states of a given system. Inexact differentials are primarily used in calculations involving heat and work because they are path functions, not state functions. Definition An inexact differential \delta u is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Differential Equation
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering. Definition Given a simply connected and open subset ''D'' of \mathbb^2 and two functions ''I'' and ''J'' which are continuous on ''D'', an implicit first-order ordinary differential equation of the form : I(x, y)\, dx + J(x, y)\, dy = 0, is called an exact differential equation if there exists a continuously differentiable function ''F'', called the potential function, so that :\frac = I and :\frac = J. An exact equation may also be presented in the following form: :I(x, y) + J(x, y) \, y'(x) = 0 where the same constraints on ''I'' and ''J'' apply for the differential equation to be exact. The nomenclature of "exact differential equation" refers to the exact differential of a function. For a function F(x_0, x_1,...,x_,x_n), the exact or total derivative with respect to x_0 is given by :\frac=\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack Exchange
Stack Exchange is a network of question-and-answer (Q&A) websites on topics in diverse fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process. The reputation system allows the sites to be self-moderating. Stack Exchange is composed of 173 communities bringing in over 100 million unique visitors each month. the three most active sites in the network are Stack Overflow (which focuses on computer programming), Mathematics, and Ask Ubuntu (focusing on the Linux distribution Ubuntu). All sites in the network are modeled after the initial site Stack Overflow which was created by Jeff Atwood and Joel Spolsky in 2008. Further Q&A sites in the network are established, defined, and eventually if found relevant brought to creation by registered users through a special site named Area 51. User contributions since May 2, 2018 are licensed under Creative Commons Attribution-ShareAlike 4.0 International. Older c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. The " =" symbol, which appears in every equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y',\ldots, y^ are the successive derivatives of the unknown function y of the variable x. Among ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
