Differential forms and smooth functions

Question

Let $f_1$ and $f_2$ two smooth functions of the real variables $x$ and $y$. My questions is about the existence of a function $g$ non-zero such that $gf_1dx+gf_2dy$ is the differential of a smooth function. Is this equivalent to find a $g$ such that $(\partial/\partial y)(gf_1)=(\partial/\partial x)(gf_2)$? If this is true, how can I prove the existence of such $g$? Thanks!

Answer 1

Hint: Existence of differential form is one of the Goal of De_Rham_cohomology which investigate for differential form with prescribed properties. and the best appraoch which presented above by Akaba comment is g(x)=0 since it is smooth function and satisfies the titled differential form .