<p>In this paper, we discuss the solutions of the nonlinear differential-difference equations <Equation ID="Equ37"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_Equ37.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="383" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \qquad f^{n}(z)+wf^{n-1}(z)f^{\prime }(z)+f^{(k)}(z+c)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z} \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mspace width="2em" /> <msup> <mi>f</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>w</mi> <msup> <mi>f</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mi>z</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mi>z</mi> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>and <Equation ID="Equ38"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_Equ38.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="449" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \qquad f^{n}(z)+wf^{n-1}(z)f^{\prime }(z)+q(z)e^{Q(z)}f^{(k)}(z+c)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2} z}, \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mspace width="2em" /> <msup> <mi>f</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>w</mi> <msup> <mi>f</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>q</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mi>z</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>α</mi> <mn>2</mn> </msub> <mi>z</mi> </mrow> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\ge 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> are integers, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">\(w, c, p_{1}, p_{2}, \alpha _{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha _{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>α</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are non-zero constants satisfying <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha _{1}\ne \alpha _{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>≠</mo> <msub> <mi>α</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_842_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(q\not \equiv 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≢</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a polynomial and <i>Q</i> is a non-constant polynomial.</p>

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On solutions of certain nonlinear differential-difference equations

  • Garima Pant

摘要

In this paper, we discuss the solutions of the nonlinear differential-difference equations \(\begin{aligned} \qquad f^{n}(z)+wf^{n-1}(z)f^{\prime }(z)+f^{(k)}(z+c)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z} \end{aligned}\) f n ( z ) + w f n - 1 ( z ) f ( z ) + f ( k ) ( z + c ) = p 1 e α 1 z + p 2 e α 2 z and \(\begin{aligned} \qquad f^{n}(z)+wf^{n-1}(z)f^{\prime }(z)+q(z)e^{Q(z)}f^{(k)}(z+c)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2} z}, \end{aligned}\) f n ( z ) + w f n - 1 ( z ) f ( z ) + q ( z ) e Q ( z ) f ( k ) ( z + c ) = p 1 e α 1 z + p 2 e α 2 z , where \(n\ge 2\) n 2 , \(k\ge 0\) k 0 are integers, \(w, c, p_{1}, p_{2}, \alpha _{1}\) w , c , p 1 , p 2 , α 1 and \(\alpha _{2}\) α 2 are non-zero constants satisfying \(\alpha _{1}\ne \alpha _{2}\) α 1 α 2 , \(q\not \equiv 0\) q 0 is a polynomial and Q is a non-constant polynomial.