I think you got caught up in some simple confusion. The Laplace Transform transforms a function of some variable (it could be x or t or whatever) to a function of s by the rule
F(s)=\int_{0}^{\infty}f(t)e^{-st}dt
So in your case f(x)\longrightarrow F(s) and your equation will go...
It is common practice to write it as
M(x,y)dx+N(x,y)dy=0
and then to differentiate N with respect to x and M with respect to y to check if the equation is exact. The whole method depends on the fact that there is some function where,
\frac{\partial \Phi}{\partial x}=M(x,y),\text{...