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Also known as: exact differential equation, total differential equation

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- Related Topics:
- differential equation

**exact equation**, type of differential equation that can be solved directly without the use of any of the special techniques in the subject. A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation. The equation *P*(*x*,*y*)*d**y*/*d**x* + *Q*(*x*,*y*)=0,or in the equivalent alternate notation *P*(*x*,*y*)*d**y* + *Q*(*x*,*y*)*d**x*=0,is exact if ∂*P*(*x*,*y*)/∂*x* = ∂*Q*(*x*,*y*)/∂*y*.In this case, there will be a function *R*(*x*,*y*), the partial *x*-derivative of which is *Q* and the partial *y*-derivative of which is *P*, such that the equation *R*(*x*,*y*)=*c* (where *c* is constant) will implicitly define a function *y* that will satisfy the original differential equation.

For example, in the equation (*x*^{2}+2*y*)*d**y*/*d**x*+2*x**y*+1=0,the partial *x*-derivative of *x*^{2}+2*y* is 2*x* and the partial *y*-derivative of 2*x**y*+1 is also 2*x*, and the function *R*=*x*^{2}*y*+*x*+*y*^{2} satisfies the conditions *R*_{x}=*Q* and *R*_{y}=*P*. The function defined implicitly by *x*^{2}*y*+*x*+*y*^{2}=*c* will solve the original equation. Sometimes if an equation is not exact, it can be made exact by multiplying each term by a suitable function called an integrating factor. For example, if the equation 3*y*+2*x**y*′ =0 is multiplied by 1/*x**y*, it becomes 3/*x*+2*y*′/*y*=0, which is the direct result of differentiating the equation in which the natural logarithmic function (ln) appears: 3ln*x*+2ln*y*=*c*, or equivalently *x*^{3}*y*^{2}=*c*, which implicitly defines a function that will satisfy the original equation.

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thermodynamics: Entropy as an exact differential

Higher-order equations are also called exact if they are the result of differentiating a lower-order equation. For example, the second-order equation *p*(*x*)*d*^{2}*y*/*d**x*^{2} +*q*(*x*)*d**y*/*d**x* +*r*(*x*)*y*=0is exact if there is a first-order expression*p*(*x*)*d**y*/*d**x*+*s*(*x*)*y*such that its derivative is the given equation. The given equation will be exact if, and only if,*p**d*^{2}*y*/*d**x*^{2} +*q**d**y*/*d**x* +*r*=0,in which case *s* in the reduced equation will equal *q*−*p*(*d**y*/*d**x*). If the equation is not exact, there may be a function *z*(*x*), also called an integrating factor, such that when the equation is multiplied by the function *z* it becomes exact.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.