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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.
Differentials and First-Order Equations
Published in L.M.B.C. Campos, Non-Linear Differential Equations and Dynamical Systems, 2019
A sixth method is (vi) a change of variable that may render a differential equation solvable; for example the homogeneous first-order differential equation, in which the dependent and independent variable appear only through their ratio, can be transformed into a separable type, and hence integrated in all cases (section 3.7). Proceeding from the first six to the next two classes, any first-order differential equation is equivalent to a first-order differential in two variables (section 3.8) leading to two cases: (vii) if it satisfies a condition of integrability, it is an exact differential that is the differential of a function, that equated to an arbitrary constant supplies the general integral; (viii) if the integrability condition is not satisfied, the first-order differential equation is equivalent to an inexact differential, and it is not the differential of a function, though it becomes so when multiplied by an integrating factor that always exists. This is no longer the case for a first-order differential in three variables (section 3.9), when there are three possibilities: (i/ii) exact (inexact) differential satisfying (not satisfying) an integrability condition and not needing (needing) an integration factor, as for a first-order differential in two variables; (ii/iii) the existence (non-existence) of an integrating factor depends on the satisfaction (non-satisfaction) of a more general integrability condition. The first-order differential may be extended to: (i) more than three variables (notes 3.1–3.15); (ii) homogeneous differentials (notes 3.16–3.20); (iii) higher-order differentials (notes 3.21–3.24).
Special, Second, and Higher-Order Equations
Published in L.M.B.C. Campos, Higher-Order Differential Equations and Elasticity, 2019
It has been shown that the four simplest thermodynamic process are (i) equivoluminar (5.93a–c), (ii) isobaric (5.95a–d); (iii) isothermal (5.96a–d), and (iv) adiabatic (5.95a–e); that is, respectively at constant (i) specific volume, (ii) pressure, (iii) temperature, and (iv) entropy or without heat exchanges. The work (5.85a) [heat (5.85b)] are: (i) generally inexact differentials, and depend on the path between two thermodynamic equilibrium states; (ii) may coincide with a function of state for a particular thermodynamic process (5.94c, d; 5.96b, c) [(5.93b, c; 5.95b, c)], thus becoming an exact differential, and independent of the path between the same initial and final equilibrium states. The heat may or may not be an exact differential depending on the thermodynamic process, but the entropy is always an exact differential given in any thermodynamic process by (5.85d)≡(5.97a) or (5.89b)≡(5.97b) that involve only functions of state: dS=dUT+pTdV=dHT−VTdp;ρV=1:dS=dUT−Pρ2Tdρ,dF=pρ2dρ−SdT.
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
In contrast, if there is no single function satisfying Equations (2), then and must be expressed in terms of two functions, say and , i.e. with . In these cases Equation (1) cannot be written as an exact differential, and is therefore described as an inexact differential equation. Inexact equations are usually more difficult to solve than exact differential equations, but can be made exact following multiplication by an appropriate integrating factor (Cortez & de Oliveira, 2017; Riley et al., 2006).