Confusion about exact DEs

Question

If I understand the definition of an exact de correctly, if $M(x,y)dx+N(x,y)dy=0 $ and $M_y=N_x$, then $$f(x,y)=\int M(x,y)dx + g(y)$$ and $$g'(y)=\int M(x,y)dx-N(x,y)$$

1) Is my interpretation correct?

2) Is M always the expression in the given equation preceding dx? Is N always the expression preceding Dy?

Answer 1

1. Your interpretation is partially correct. $$ F(x, y) := \int M(x,y)dx + g(y) $$ However: $$ g'(y) := N(x,y) - \frac{\partial}{\partial y} \int M(x,y)dx $$ This is because $\frac{\partial F}{\partial y} = N$. $$ F(x, y) := \int M(x,y)dx + g(y) \implies N(x, y) = \frac{\partial}{\partial y}\int M(x,y)dx + \frac{\partial}{\partial y}g(y) \\ = \frac{\partial}{\partial y}\int M(x,y)dx + g'(y) $$

2. Yes, $M(x, y)$ precedes $dx$ and $N(x, y)$ precedes $dy$