= Integrating factor depending on one coordinate
{title2=$\mu'/\mu=(P_y-Q_x)/Q$}
For a <differential one-form> $P\,dx+Q\,dy$, seek a nonzero <integrating factor> $\mu(x)$. Expanding $(\mu P)_y=(\mu Q)_x$ gives the displayed equation where $Q\ne0$. 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 $\mu'/\mu=(Q_x-P_y)/P$ where $P\ne0$. The resulting product is an <exact differential form> on a sufficiently small regular domain; singular lines of the factor need separate treatment.
Back to article page
