关于这个微分方程的求解

葡萄糖

与椭圆积分有关的对称配方表达式


\begin{gather*}
{\large{\int}}_{u}^{+\infty}\dfrac{\mathrm{d}t}{\sqrt{1+\alpha\,\!t+\beta\,\!t^2+\gamma\,\!t^3}}+{\large{\int}}_{v}^{+\infty}\dfrac{\mathrm{d}t}{\sqrt{1+\alpha\,\!t+\beta\,\!t^2+\gamma\,\!t^3}}={\large{\int}}_{w}^{+\infty}\dfrac{\mathrm{d}t}{\sqrt{1+\alpha\,\!t+\beta\,\!t^2+\gamma\,\!t^3}} \\
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w=\dfrac{1}{\gamma}\left(\frac{\sqrt{1+\alpha\,\!u+\beta\,\!u^2+\gamma  u^3}-\sqrt{1+\alpha\,\!v+\beta\,\!v^2+\gamma\,\!v^3}}{u-v}\right)^2-u-v-\frac{\beta}{\gamma}\\
\\
\Downarrow\\   
\\
[\alpha -\gamma\left(uv+uw+vw\right)]^2=4 (1+\gamma\,\!uvw)[\beta+\gamma\,\!(u+v+w)]
\end{gather*}

葡萄糖

与超椭圆积分有关的对称配方表达式


\begin{gather*}
{\large{\int}}_0^{u}\dfrac{\mathrm{d}t}{\sqrt{(1-a^2t^2)(1-b^2t^2)(1-c^2t^2)}}+{\large{\int}}_0^{v}\dfrac{\mathrm{d}t}{\sqrt{(1-a^2t^2)(1-b^2t^2)(1-c^2t^2)}}\\={\large{\int}}_0^{w}\dfrac{\mathrm{d}t}{\sqrt{(1-a^2t^2)(1-b^2t^2)(1-c^2t^2)}} \\
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\big[u^2+v^2+w^2-(a^2 b^2+a^2 c^2+b^2 c^2)u^2v^2w^2\big]^2\\
=4 \left[u^2 v^2+u^2 w^2+v^2 w^2-(a^2+b^2 +c^2)u^2 v^2 w^2\right]\left(1-a^2 b^2 c^2 u^2 v^2 w^2\right)
\end{gather*}

葡萄糖

\begin{gather*}
\int_{0}^{u}\frac{\mathrm{d}t}{\sqrt{\lambda+\mu t^{2}+\nu t^{4}}}+\int_{0}^{v}\frac{\mathrm{d}t}{\sqrt{\lambda+\mu t^{2}+\nu t^{4}}}=\int_{0}^{w}\frac{\mathrm{d}t}{\sqrt{\lambda+\mu t^{2}+\nu t^{4}}}\\
\\
w=\frac{u\sqrt{\lambda\left(\lambda+\mu\>\!v^{2}+\nu\>\!v^{4}\right)}+v\sqrt{\lambda\left(\lambda+\mu\>\!u^{2}+\nu\>\!u^{4}\right)}}{\lambda-\nu\,u^{2}v^{2}}\\
\\
\lambda\left(u^{2}+v^{2}\right)+2uv\sqrt{\lambda\left(\lambda+\mu\>\!w^{2}+\nu\>\!w^{4}\right)}=\left(\lambda+\nu\>\!u^{2}v^{2}\right)w^{2}\\
\\
\Big[\lambda  \left(u^2+v^2+w^2\right)-\nu \>\!u^2 v^2 w^2\Big]^2=4 \lambda  \Big[\lambda\left(u^2 v^2+u^2 w^2+v^2 w^2\right)+\mu\>\!u^2 v^2 w^2\Big]
\end{gather*}

\begin{gather*}
\int_{0}^{u}\frac{\mathrm{d}t}{\sqrt{\alpha\>\!t+\beta\>\!t^{2}+\gamma\>\!t^{3}+\delta\>\!t^{4}}}+\int_{0}^{v}\frac{\mathrm{d}t}{\sqrt{\alpha\>\!t+\beta\>\!t^{2}+\gamma\>\!t^{3}+\delta\>\!t^{4}}}=\int_{0}^{w}\frac{\mathrm{d}t}{\sqrt{\alpha\>\!t+\beta\>\!t^{2}+\gamma\>\!t^{3}+\delta\>\!t^{4}}}\\
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w=\frac{\left[\sqrt{\alpha\>\!u \left(\alpha+\beta\>\!u+\gamma\>\!u^2+\delta\>\!u^3\right)}+\sqrt{\alpha\>\!v \left(\alpha+\beta\>\!v+\gamma\>\!v^2+\delta\>\!v^3\right)}\right]^2-\alpha  (u-v)^2 \left(\beta +\gamma(u+v)+\delta  (u+v)^2\right)}{(\alpha -\gamma\>\!uv)^2-4 \delta\>\!uv[\alpha(u+v)+\beta\>\!uv]}\\
\\
\big[\alpha(u+v+w)-\gamma\>\!uvw\big]^2=4\big[\alpha(uv+uw+vw)+\beta\>\!uvw\big]\big(\alpha + \delta\>\!uvw\big)
\end{gather*}