In this article, we study the meromorphic solutions of the following non-linear differential equation<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_321_Article_Equa.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="212" /> </MediaObject> <EquationSource Format="TEX">$f^{n}(z)+P_{d}(z,f)=\sum_{j=1}^{k}p_{j}{\rm e}^{\alpha_{j}z},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>f</mi> <mrow> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>P</mi> <mrow> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow> <mi>j</mi> </mrow> </msub> <msup> <mrow> <mi mathvariant="normal">e</mi> </mrow> <mrow> <msub> <mi>α</mi> <mrow> <mi>j</mi> </mrow> </msub> <mi>z</mi> </mrow> </msup> <mo>,</mo> </math></EquationSource> </Equation>where n and k are integers with n ≥ k ≥ 3, Pd(z, f) is a differential polynomial in f of degree d ≤ n − 1, p′js and α′js are non-zero constants. We obtain the expressions of meromorphic solutions of the above equations under some restrictions on α′js. Some examples are given to illustrate the possibilities of our results.

Meromorphic solutions of certain types of differential equations with k exponential terms

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发表日期：

2025年3月11日

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Minfeng Chen,
Meili Liang,
Zongsheng Gao

摘要

In this article, we study the meromorphic solutions of the following non-linear differential equation

$f^{n}(z)+P_{d}(z,f)=\sum_{j=1}^{k}p_{j}{\rm e}^{\alpha_{j}z},$ $f^{n} (z) + P_{d} (z, f) = \sum_{j = 1}^{k} p_{j} e^{α_{j} z},$

where n and k are integers with n ≥ k ≥ 3, P_d(z, f) is a differential polynomial in f of degree d ≤ n − 1, p′_js and α′_js are non-zero constants. We obtain the expressions of meromorphic solutions of the above equations under some restrictions on α′_js. Some examples are given to illustrate the possibilities of our results.