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Entropy
Published in Kavati Venkateswarlu, Engineering Thermodynamics, 2020
However, since (δQ/T)rev is a thermodynamic property, it is an exact differential. From a mathematical perspective, an integrating factor is required to convert an inexact differential to an exact differential. 1/T serves as the integrating factor in converting the inexact differential δQ to the exact differential δQ/T for a reversible process.
Work, Power, and Energy Applied to the Dynamics of Particles
Published in G. Boothroyd, C. Poli, Applied Engineering Mechanics, 2018
The forces involved if Eq. (15.24) applies are called conservative forces, and mechanical energy has been conserved. The forces involved if Eq. (15.23) applies are nonconservative forces, and the work done has been converted into some nonmechanical form of energy. When the integral of a function around a closed path is zero, the function is an exact differential, and we can write, for conservative forces only, () Fxdx+Fydy=−dV where V is called the potential energy and the negative sign indicates that potential energy is a measure of the capacity for doing work.
Traditional First-Order Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
is called an exact differential equation. As long as x = C (a constant) is not a solution, we consider the equivalent form () M(x,y)+N(x,y)dydx=0
An elementary proof by contradiction of the exactness condition for first-order ordinary differential equations
Published in International Journal of Mathematical Education in Science and Technology, 2021
When written in terms of differentials, the general form of the first-order ordinary differential equation is (Arfken & Weber, 2005; Boyce & DiPrima, 2009; Cortez & de Oliveira, 2017; Glynn, 2001; Kreyszig, 2006; Piskunov, 1974/1996; Riley et al., 2006; Robinson, 2004) where and are functions of and . If some function can be found such that then Equation (1) is described as an exact differential equation because its left-hand-side can then be expressed in terms of the exact (or total) differential of , viz. Exact differential equations can be integrated to yield the implicit solution
Lie group method for constructing integrating factors of first-order ordinary differential equations
Published in International Journal of Mathematical Education in Science and Technology, 2023
An integrating factor, first proposed by Clairaut in 1739, can transform a first-order ordinary differential equation which is not an exact differential form, into an exact differential form. However, it is difficult to find an integrating factor of the first-order ordinary differential equation, because the acquisition of integrating factors is usually more complicated than the integration of the original equation (Ibragimov, 2013; Simmons, 1991).