How do I solve this equation in terms of Bessel's functions? $xy'' + y = 0, \ x > 0$

Question

My current progress is using the standard equation $xy'' + cy' + k^2(x^r) y = 0$, and I get $c = 0, r = 0, m = 1, a = 1/2$. But I am not sure what my next step should be or whether this is correct. Could anyone kindly help me please?

Answer 1

The solution of your equation is $$c_1 \sqrt{x} J_1\left(2 \sqrt{x}\right)+2 i c_2 \sqrt{x} Y_1\left(2 \sqrt{x}\right).$$ In order to get the standard Bessel equation with a linear argument, its evident that you have to substitute the independent variable $x=w^2$ with $$\partial_{x,x}y(x^2) = 2 y'(x^2) +4 x^2 \ y''(x^2)$$ The number of different forms of linear ODE's of second order yielding Bessel functions as solutions is two pages with 34 entries in Gradshteyn/Rhyzik 8.49