= Monomial integrating factor
{title2=$\mu(x,y)=x^\alpha y^\beta$}
For a <differential one-form> $M\,dx+N\,dy$, a monomial <integrating factor> exists on a chosen domain away from the axes when constants $\alpha,\beta$ make
$$
M_y-N_x=\frac{\alpha}{x}N-\frac{\beta}{y}M.
$$
This 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 $dF=\mu M\,dx+\mu N\,dy$ 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.
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