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Identifying first order exact differential equations. Let the functions , , , and , where the subscripts denote the partial derivative with respect to the relative variable, be continuous in the region . Then the differential equation. is exact if and only if. That is, there exists a function , called a potential function, such that.
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 path-independent, and this fact is used to identify state functions in thermodynamics .
An ordinary differential equation ( ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. The unknown function is generally represented by a variable (often denoted y ), which, therefore, depends on x. Thus x is often called the independent variable of the equation.
In mathematics, especially vector calculusand differential topology, a closed formis a differential formαwhose exterior derivativeis zero (dα= 0), and an exact formis a differential form, α, that is the exterior derivative of another differential form β. Thus, an exactform is in the imageof d, and a closedform is in the kernelof d.
The exact solution of the differential equation is () =, so () =. Although the approximation of the Euler method was not very precise in this specific case, particularly due to a large value step size h {\displaystyle h} , its behaviour is qualitatively correct as the figure shows.
An inexact differential equation is a differential equation of the form (see also: inexact differential ) {\displaystyle M (x,y)\,dx+N (x,y)\,dy=0, {\text { where }} {\frac {\partial M} {\partial y}} eq {\frac {\partial N} {\partial x}}.} The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739 ...
In mathematics, an ordinary differential equation ( ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown (s) consists of one (or more) function (s) and involves the derivatives of those functions. [ 1] The term "ordinary" is used in contrast with partial differential equations (PDEs ...
Riccati equation. In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form. where and . If the equation reduces to a Bernoulli equation, while if the equation becomes a first order linear ordinary ...
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