Monomial integrating factor ($\mu (x,y)=x^{\alpha }{y}^{\beta }$)

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.