Let R be a prime ring of characteristic different from 2, U be its Utumi quotient ring with extended centroid C and f(x1,…,xn)\documentclass[12pt]{minimal}
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\begin{document}$$f(x_1,\ldots ,x_n)$$\end{document} be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations on R such that F2(f(r))f(r)-G(f(r)2)=0\documentclass[12pt]{minimal}
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\begin{document}$$F^2(f(r))f(r)-G(f(r)^2)=0$$\end{document} for all r=(r1,…,rn)∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$r=(r_1,\ldots ,r_n)\in R^{n}$$\end{document}. Then one of the following holds:there exists a∈U\documentclass[12pt]{minimal}
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\begin{document}$$a\in U$$\end{document} such that F(x)=xa\documentclass[12pt]{minimal}
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\begin{document}$$F(x)=xa$$\end{document} and G(x)=a2x\documentclass[12pt]{minimal}
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\begin{document}$$G(x)=a^2x$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document} with a2∈C\documentclass[12pt]{minimal}
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\begin{document}$$a^2\in C$$\end{document};there exists a∈U\documentclass[12pt]{minimal}
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\begin{document}$$a\in U$$\end{document} such that F(x)=ax\documentclass[12pt]{minimal}
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\begin{document}$$F(x)=ax$$\end{document} and G(x)=a2x\documentclass[12pt]{minimal}
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\begin{document}$$G(x)=a^2x$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document};f(r1,…,rn)2\documentclass[12pt]{minimal}
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\begin{document}$$f(r_1,\ldots ,r_n)^2$$\end{document} is central valued on R and one of the following holds:there exist a,b∈U\documentclass[12pt]{minimal}
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\begin{document}$$a, b\in U$$\end{document} such that F(x)=xa\documentclass[12pt]{minimal}
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\begin{document}$$F(x)=xa$$\end{document} and G(x)=xa2+[b,x]\documentclass[12pt]{minimal}
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\begin{document}$$G(x)=xa^2+[b,x]$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document} with a2∈C\documentclass[12pt]{minimal}
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\begin{document}$$a^2\in C$$\end{document};there exist a,b∈U\documentclass[12pt]{minimal}
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\begin{document}$$a,b\in U$$\end{document} such that F(x)=ax\documentclass[12pt]{minimal}
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\begin{document}$$F(x)=ax$$\end{document} and G(x)=xa2+[b,x]\documentclass[12pt]{minimal}
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\begin{document}$$G(x)=xa^2+[b,x]$$\end{document} for all x∈R\documentclass[12pt]{minimal}
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\begin{document}$$x\in R$$\end{document}.
