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Differential integrating factor

In the example of the previous section, the release of the stop always leads to the motion of the piston in one direction, to a final state in which the pressures are equal, never in the other direction. This obvious experimental observation turns out to be related to a mathematical problem, the integrability of differentials in themiodynamics. The differential Dq, even is inexact, but in mathematics many such expressions can be converted into exact differentials with the aid of an integrating factor. [Pg.333]

Equation 3-133 is a first order linear differential equation of the form dy/dx -i- Py = Q. The integrating factor is IF = and... [Pg.141]

When the Pfaffian expression is inexact but integrable, then an integrating factor A exists such that AbQ = d5, where dS is an exact differential and the solution surfaces are S = constant. While solution surfaces do not exist for the inexact differential 8Q, solution curves do exist. The solution curves to dS = 0 will also be solution curves to bQ = 0. Since solution curves for dS on one surface do not intersect those on another surface, a solution curve for 8Q — 0 that lies on one surface cannot intersect another solution curve for bQ = 0 that lies on a different surface. [Pg.611]

For the two-dimensional Pfaffian, condition (c) above is no longer applicable. That is, the Pfaffian differential expression is either (a), exact, or (b), integrable with an integrating factor. [Pg.611]

Schottky effect in solids 580-5 Second law of thermodynamics 56-90 absolute temperature, identification of as integrating factor 71-8 Caratheodory principles differentials 63-7 and inaccessible states 68-71... [Pg.661]

We can find the solution to Eq. (4-1), which is simply a set of first order differential equations. As analogous to how Eq. (2-3) on page 2-2 was obtained, we now use the matrix exponential function as the integration factor, and the result is (hints in the Review Problems)... [Pg.77]

The differential equations (36) and (37) are solved by the integrating factor method, with the boundary condition C = C0 at x = 0. [Pg.260]

This first-order linear differential equation may be solved using an integrating factor approach to give... [Pg.149]

Finally, we briefly mention the concept of an integrating factor, a multiplicative factor (L) that converts an inexact differential (ctf) to an exact differential (dg), namely,... [Pg.16]

It was initially appreciated by R. Clausius that Carnot s theorem (4.25) allows the second law to be reformulated in a profoundly improved form. Clausius recognized that (4.25) is nothing more than the exactness condition (1.16a) for the differential dqmY/Ti.e., that L = 1 /T is an integrating factor for the inexact differential state property, a conserved quantity that... [Pg.136]

Use the integrating factor method to find the general solutions to first-order differential equations linear in y... [Pg.136]

There are thus two integrations to perform one to determine the integrating factor, and a second which involves the product R(x)Q x) as integrand. Since we are dealing with a first-order differential equation, we expect only one constant of integration, but, from the above discussion, it appears that two such constants may arise. We now describe why there is, in fact, only one undetermined constant of integration. [Pg.145]

Finding general solutions to linear first order differential equations using the integrating factor method. [Pg.158]

This is a first-order linear differential equation and can be solved using the method of integrating factors that we show below. Multiplying both sides by e1 2, we have... [Pg.67]

At the lime the article was written, most physicists were still under the spell of the derivation by Clausius of the second law of thermodynamics in the form of the existence of an integrating factor for the well-known expression for the quantity of heat AQ put into the system. In this derivation the irreversibility in time of all processes occurring in nature played an important role. Hence it seemed that the possibility of a reversal of the natural development (which according to the Wieder-kehreinwand of Zermelo should occur after a sufficiently long time) threatened the validity of some of the most important results of thermodynamics. However, it became clear to me afterwards, that the existence of an integrating factor has to do only with the mathematical expression of AQ=dU+dA in terms of the differentials dxi, dxj, , dx of the equilibrium parameters Xi,... [Pg.139]

We see, then, that if an integrating factor 1/q exists that converts the Pfaffian form HL, into an exact differential, Eq. (1.11.7), then the coefficients X in the relation 3L -X]dx2 + X2dx2 + X3dx3 must obey (1.11.12). The latter relation may readily be generalized to the more general case n > 3, by replacing the subscripts 1, 2, 3 in Eq. (1.11.12) with i, j, k, respectively. [Pg.74]

If a function L does not admit of an exact differential of the form (9.1.2) it may nevertheless be possible, under conditions established below, to set up functions q x, ..., Xi,..., Xn) such that the ratio dL/q = dR does constitute an exact differential. Pfaffian forms of this genre are of special interest they are said to be holonomic or integrable. For obvious reasons q is said to be an integrating denominator and 1/, an integrating factor. [Pg.428]

If Xi dxi is not an exact differential but is nevertheless holonomic then it has an associated integrating factor such that qdR = J2i dxi, whence... [Pg.428]

We see then that if an integrating factor /q exists that converts the Pfaf-fian dL into the exact differential (9.2.7), then the coefficients in the relation dL = X dx + X2dx2 + X2,dx3 are subject to the requirement (9.2.12). For the more general case of n > 3 we obtain very similar relations, with 1, 2, 3 in (9.2.12) being replaced by indices i, j, k that are cyclically permuted, in the manner shown in Eq. (9.2.20) below. [Pg.431]

I>( c) is sometimes referred to as being an integrating factor, used to convert equation (2.1) to an exact differential equation. ... [Pg.26]

The solution to this linear differential equation gives the pounds of Na2S04 after 10 min the concentration then can be calculated as jc/y. To integrate the linear differential equation, we introduce the integrating factor... [Pg.641]

The solution of this first-order differential equation may be obtained through use of an integrating factor. The solution is... [Pg.249]

Moreover, as pointed out in Section 1.3, dexact differential such that the factor 1/ (A2T) in Eq. (2.53) must be an integrating factor because dw cannot be expected to be an exact differential per se. Hence, we can write... [Pg.53]

This is a first-order linear differential equation that can be solved analytically by using an integrating factor. We solve (1) subject to the initial condition that (0) = 0, and obtain... [Pg.206]


See other pages where Differential integrating factor is mentioned: [Pg.48]    [Pg.48]    [Pg.409]    [Pg.22]    [Pg.71]    [Pg.49]    [Pg.49]    [Pg.244]    [Pg.16]    [Pg.146]    [Pg.141]    [Pg.409]    [Pg.715]    [Pg.245]    [Pg.15]    [Pg.14]    [Pg.70]    [Pg.75]    [Pg.77]    [Pg.431]    [Pg.436]    [Pg.271]   
See also in sourсe #XX -- [ Pg.282 , Pg.284 ]




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