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<p>In this paper, we present a necessary and sufficient condition for the existence of rational first integrals of the following separable differential equation: <Equation ID="Equ12"> <EquationSource Format="TEX">\(\begin{aligned} \frac{dy}{dx}=f(x)g(y) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfrac> <mrow> <mi mathvariant="italic">dy</mi> </mrow> <mrow> <mi mathvariant="italic">dx</mi> </mrow> </mfrac> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <i>f</i>(<i>x</i>),&#xa0;<i>g</i>(<i>y</i>) are two univariate rational functions. We also present an algorithm to verify the condition and to compute a rational first integral when the condition is satisfied.</p>

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Rational First Integrals of Separable Differential Equations

  • 期刊论文

  • 发表日期：

    2025年4月19日

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  • Feng Ruyong,
  • Guo Zewang,
  • Xiong Siting

摘要

In this paper, we present a necessary and sufficient condition for the existence of rational first integrals of the following separable differential equation: \(\begin{aligned} \frac{dy}{dx}=f(x)g(y) \end{aligned}\) dy dx = f ( x ) g ( y ) where f(x), g(y) are two univariate rational functions. We also present an algorithm to verify the condition and to compute a rational first integral when the condition is satisfied.