Transforming $x^2y''(x) +Axy' (x) +Bx^{\delta}y(x)=0$ into bessel-function

Question

Given the equation :

$x^2y''(x) +Axy' (x) +Bx^{\delta}y(x)=0$

Need to transform it to bessel-function.

My attempt:

I substitute :$x=z^{\alpha}$ so $(z^{\alpha})^2y''(x)+A(z^{\alpha}) y' (x)+B(z^{\alpha}) ^{\delta}y(x)=0$

But I'm having difficulty to write it in terms of $y(z)$ especially the $y''(x)$ term.

How do I do that? Thank you.

Answer 1

The correct substitution is $$ y(x)=x^{\frac{1}{2}-\frac{A}{2}} f\left(\frac{2 \sqrt{B} x^{\delta /2}}{\delta }\right) $$ This transforms your equation to the Bessel one for $f(z)$ with the index $\nu=\frac{A-1}{\delta }$.