Para acharmos o fator integrante, usaremos os cálculos abaixo:

\(\begin{align} & u=u(x,y) \\ & u(x,y)u(x,y)+u({{x}_{0}},y)N(x,y)y'=0 \\ & u(x,y)u(x,y)+u({{x}_{0}},y)N(x,y)y'=\frac{d}{dy}+\frac{d}{dx} \\ & {{u}_{y}}M+{{M}_{y}}u={{u}_{x}}N+{{N}_{x}}u \\ & \left[ \frac{d}{dy}u(x,y) \right]M(x,y)+\left[ \frac{d}{dy}M(x,y) \right]={{M}_{oy}} \\ & \left[ \frac{d}{dy}u(x,y) \right]M(x,y)+\left[ \frac{d}{dy}M(x,y) \right]=={{N}_{ox}} \\ & {{M}_{y}}u=u'N+u{{N}_{x}} \\ \end{align} \)

Portanto, o fator integrante será de \(\boxed{{M_y}u = u'N + u{N_x}}\).