Integrating factor for a differential one-form (${\partial }_x(\mu f)={\partial }_y(\mu g)$)

For the differential one-form $fdy+gdx$, a nonzero integrating factor $\mu$ makes $\mu fdy+\mu gdx$ an exact differential form locally. If it equals $dF$, then $F_y=\mu f$, $F_x=\mu g$, and equality of mixed partial derivatives gives the displayed compatibility condition. Conversely this condition is local sufficiency for smooth coefficients on a small rectangle; global exactness additionally depends on the domain. Zeros of the multiplying factor can change the solution set, so equivalence is asserted only where the factor is nonzero.

For a differential one-form $Pdx+Qdy$, seek a nonzero integrating factor $\mu (x)$. Expanding $(\mu P{)}_y=(\mu Q{)}_x$ gives the displayed equation where $Q\ne 0$. Such a factor exists locally when $(P_y-Q_x)/Q$ depends only on $x$, and is $\mu =\exp \int [(P_y-Q_x)/Q]dx$. Likewise, a factor depending only on $y$ obeys ${\mu }^{\prime }/\mu =(Q_x-P_y)/P$ where $P\ne 0$. The resulting product is an exact differential form on a sufficiently small regular domain; singular lines of the factor need separate treatment.

For a differential one-form $Mdx+Ndy$, a monomial integrating factor exists on a chosen domain away from the axes when constants $\alpha ,\beta$ makeThis follows by expanding the exactness condition ${\partial }_y(\mu M)={\partial }_x(\mu N)$. When $M,N$ are sums of monomials, matching their coefficients often reduces the search for $\alpha ,\beta$ to a small linear system. Integrating $dF=\mu Mdx+\mu Ndy$ then gives the implicit solution $F=C$. Curves excluded by the multiplying factor, especially solutions on coordinate axes, must be tested in the original equation.