Meromorphic solutions of the differential equation $(f^{n})^{(k)}(g^{n})^{(k)}=1$ and its applications

In 1997, C. C. Yang and X. H. Hua [Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406] proved that if $f$ and $g$ are two non-constant meromorphic functions such that $f^nf'g^n g'= 1,$ where $n$ is a positive integer satisfying $n\geq 6,$ then $f$ and $g$ are