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I am having a difficulty solving this second order linear ODE with a variable coefficient. Below is the equation:

$$x^\beta\,y'' - \beta\,x^{\beta-1}y' - \gamma\,y = 0 $$

With the following boundary conditions $ y(0) = 0 $ and $y(L) = 0$. Where $ \gamma $ is a constant. I have employed the series method of solution, the Frobenius method to be precise. I also tried to transform the equation to a Bessel kind or something more familiar, but I have been unsuccessful. I tried the Sturm-Liouville approach for solving Second ODE.

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    One can wrangle this into the form of a modified Bessel equation with a couple of substitutions. I'm not sure if that's quite what you want, but for a general $\beta$, one can't do any better. – 2017-02-28
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    Actually am looking at a general case of $\beta$ hmmmmm – 2017-02-28

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Let $$\xi=\int\frac{1}{x^{\beta/2}}dx=\frac{1}{(1-\beta/2)x^{\beta/2-1}}$$ the equation becomes $$\xi\frac{d^{2}}{d\xi^{2}}y(\xi)-\frac{3\beta}{2-\beta}\frac{d}{d\xi}y(\xi)-\gamma\xi{y(\xi)}=0$$ And the solution is $$y(\xi)=c_{1}\xi^{\frac{1+\beta}{2-\beta}}J_{\frac{\beta+1}{\beta-2}}(-i\sqrt{\gamma}\xi)+c_{2}\xi^{\frac{1+\beta}{2-\beta}}Y_{\frac{\beta+1}{\beta-2}}(-i\sqrt{\gamma}\xi)$$