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Find the general solution of $$\frac{{{\partial }^{2}}W}{\partial {{\xi }^{2}}}=\left( W+\frac{1}{W}\cdot \frac{\partial W}{\partial \xi }-\frac{\lambda }{W} \right)\frac{\partial W}{\partial \xi }-1$$

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  • $\begingroup$ Any ideas on this? $\endgroup$ – Ryan Feb 8 '13 at 13:59
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Let $X=\dfrac{\partial W}{\partial\xi}$ ,

Then $\dfrac{\partial^2W}{\partial\xi^2}=\dfrac{\partial X}{\partial\xi}=\dfrac{\partial X}{\partial W}\dfrac{\partial W}{\partial\xi}=X\dfrac{\partial X}{\partial W}$

$\therefore X\dfrac{\partial X}{\partial W}=\left(W+\dfrac{X}{W}-\dfrac{\lambda}{W}\right)X-1$

$X\dfrac{\partial X}{\partial W}=\dfrac{X^2}{W}+\left(W-\dfrac{\lambda}{W}\right)X-1$

This belongs to an Abel equation of the second kind.

Let $X=WY$ ,

Then $\dfrac{\partial X}{\partial W}=W\dfrac{\partial Y}{\partial W}+Y$

$\therefore WY\left(W\dfrac{\partial Y}{\partial W}+Y\right)=WY^2+\left(W-\dfrac{\lambda}{W}\right)WY-1$

$W^2Y\dfrac{\partial Y}{\partial W}+WY^2=WY^2+\left(W-\dfrac{\lambda}{W}\right)WY-1$

$W^2Y\dfrac{\partial Y}{\partial W}=\left(W-\dfrac{\lambda}{W}\right)WY-1$

$Y\dfrac{\partial Y}{\partial W}=\left(1-\dfrac{\lambda}{W^2}\right)Y-\dfrac{1}{W^2}$

For the special case $\lambda=0$ , this exactly belongs to the ODE of the form http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=136.

The general solution is $\begin{cases}W=\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\\Y=\dfrac{2\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2-6\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}{\sqrt[3]{36}\tau^\frac{2}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\end{cases}$

$\begin{cases}W=\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\\X=\dfrac{2\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2-6\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^3-6\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)}{\sqrt[3]{36}\tau^\frac{2}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}W\end{cases}$

$\begin{cases}W=\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\\\dfrac{\partial W}{\partial\xi}=\dfrac{2\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3-6\tau^\frac{2}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^3-6\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)}{3\sqrt[3]{6}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)^2}\end{cases}$

$\begin{cases}W=\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\\\dfrac{\dfrac{\partial W}{\partial\tau}}{\dfrac{\partial \xi}{\partial\tau}}=\dfrac{2\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3-6\tau^\frac{2}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^3-6\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)}{3\sqrt[3]{6}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)^2}\end{cases}$

$\begin{cases}W=\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\\\dfrac{\partial \xi}{\partial\tau}=\dfrac{3\sqrt[3]{6}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)^2}{2\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3-6\tau^\frac{2}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^3-6\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)}\dfrac{\partial}{\partial\tau}\left(\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\right)\end{cases}$

$\begin{cases}W=\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\\\xi=\int^\tau\dfrac{3\sqrt[3]{6}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)^2}{2\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3-6\tau^\frac{2}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^3-6\tau^\frac{8}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^3\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)}\dfrac{\partial}{\partial\tau}\left(\dfrac{2\tau^\frac{4}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2}{\sqrt[3]{36}\left(\left(\dfrac{\tau}{2}\left(I_{-\frac{2}{3}}(\tau)+I_\frac{4}{3}(\tau)+CI_{-\frac{4}{3}}(\tau)+CI_\frac{2}{3}(\tau)\right)+\dfrac{1}{3}\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)\right)^2+\tau^2\left(I_\frac{1}{3}(\tau)+CI_{-\frac{1}{3}}(\tau)\right)^2\right)}\right)~d\tau+c\end{cases}$

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