 Solution

Exactness after multiplication requires ${\partial }_y(\mu M)={\partial }_x(\mu N)$. Since $\mu$ depends only on $x$,For $\mu =\exp (\int f(x)dx)$, the ratio ${\mu }^{\prime }/\mu$ is $f(x)$, soThe assumed existence means this ratio is a function only of $x$ wherever the division is valid. For a factor depending only on $y$, the corresponding equation is ${\psi }^{\prime }M+\psi M_y=\psi N_x$, givingThese are the integrating factor for a differential one-form formulas. Nonzero constant multiples are equivalent. Exactness follows locally from closedness, and globally on a suitable simply connected domain.