I wish to solve the following homogeneous ODE :
$y''(x) - \beta x^{-1/4}y(x)=0$
From what I've read in a couple paper where a similar equation is solved for $\beta=1$, a solution should be
$ y(x)=A\left[2\left( \frac{4}{7} \right)^{4/7}/\Gamma(\frac{4}{7}) \right]K_{4/7}\left(\frac{8}{7}x^{7/8}\right)$
where A is a constant, $\beta=1$, and where they have used the condition $y(\infty) \to 0$ to eliminate the second constant. $K_n(z)$ is the modified Bessel function of the second kind.
Using Bowman (1958) relation, I get
$y(x)=\sqrt{x}\left[ (-1)^{2/7}A_1 I_{4/7}(\sqrt{\beta} x^{7/8}) - (-1)^{5/7} A_2 I_{-4/7}(\sqrt{\beta} x^{7/8}) \right]$
where $I_n(z)$ is the modified Bessel function of the first kind. This may be incorrect, but I guess that the solution I'm looking for looks like $y(x)\sim K_{4/7}\left(\sqrt{\beta}\frac{8}{7}x^{7/8}\right)$...
Any help would be appreciated!
Thanks