## Solutions of a pair of differential equations and their applications.(English)Zbl 1062.34099

The author considers the common solutions of a pair of differential equations and gives some of their applications in the uniqueness problems of entire functions.
Let $$\alpha(z)$$ and $$\beta(z)$$ are nonconstant entire functions such that $$e^{\alpha(z)- \beta(z)}\not\equiv 1$$. Then the pair of differential equations $f^{(n)}- e^{\alpha(z)}f= 1,\quad f^{(n+ 1)}- e^{\beta(z)} f= 1,$ has no common solution.
Let $$\alpha(z)$$ be an entire function, and let $$\beta(z)$$ be a nonconstant entire function. Then the pair of differential equations $f^{(n)}- e^{\alpha(z)} f= 1,\quad f'- e^{\beta(z)} f=1,$ has no common solution. Applying these results, the author obtains: Let $$f$$ be a nonconstant entire function, $$n$$ be a positive integer. If $$f$$, $$f^{(n)}$$, and $$f^{(n+1)}$$ share a finite value $$a\neq 0$$, then $$f$$ must be of finite order.

### MSC:

 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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### References:

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