Generalized differential identities on prime rings and algebras

The goal of this study is to bring out the following conclusion. Let $R$ be a noncommutative prime ring with $2(m+n)!$ torsion freeness and let $m$ and $n$ be fixed, non-negative integers and $d,g$ be Jordan derivations on $R$. If $x^{m+n}d(x)+x^{m}g(x)x^n\in Z(R)$ or $d(x)x^{m+n}+x^{m}g(x)x^n\in Z(R)$ or $x^{n}d(x)x^m+x^{m}g(x)x^n\in Z(R)$ then $d=g=0$ follows for every $x\in R$.