Consider the problem
$$\xi^4 v'' + \alpha \, \mathrm{e}^v = 0, \quad v = v(\xi), \quad 1 < \xi < \infty, $$
with boundary conditions $v(1) = 0$ and $|v| < \infty$ for all $\xi$ and $\alpha > 0$ is a real constant. This is closely related to the so-called Bratu-Gelfand problem, but in a different geometry. I arrived at this equation after performing some change of variable on the original problem. Though this seems easier, it is most likely this problem has not a closed-form solution.
Does anyone spot a useful change of variables here? Moreover, does it have a name?
Any thoughts are greatly appreciated.
$\xi^4v''+\alpha e^v=0$
$\dfrac{d^2v}{d\xi^2}=-\alpha\xi^{-4}e^v$
Approach $1$:
Follow the method in http://science.fire.ustc.edu.cn/download/download1/book%5Cmathematics%5CHandbook%20of%20Exact%20Solutions%20for%20Ordinary%20Differential%20EquationsSecond%20Edition%5Cc2972_fm.pdf#page=440:
$\therefore\dfrac{d^2\xi}{dv^2}=\alpha\xi^{-4}e^v\left(\dfrac{d\xi}{dv}\right)^3$
Let $\begin{cases}t=\dfrac{d\xi}{dv}\\u=\xi^{-3}\end{cases}$ ,
Then $\dfrac{d^2v}{dt^2}=\dfrac{tv^{-\frac{4}{3}}}{3}\left(\dfrac{dv}{dt}\right)^2$
Which reduces to a generalized Emden–Fowler equation.
Approach $2$:
Follow the method in http://eqworld.ipmnet.ru/en/solutions/ode/ode0314.pdf:
Let $\begin{cases}t=\xi^{-2}e^v\\u=\xi\dfrac{dv}{d\xi}\end{cases}$ ,
Then $t(u-2)\dfrac{du}{dt}=-\alpha t+u$
Which reduces to an Abel equation of the second kind.