Can the number of differential equations of an expression be more than one?

Question

If we have an differential equation Mdx + Ndy=0, then will the solution obtained from it f(x,y) when operated as d(f(x,y)) give the equation same as the initial differential equation.

Answer 1

Yes, by definition $$ d(f(x,y))=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy $$

and $f(x,y)$ is a solution of $Mdx+Ndx=0$ if $M=\frac{\partial f}{\partial x}$ and $N=\frac{\partial f}{\partial y}$