I have a Bessel function in standard form but with a rational part on the right $ x^2\frac{d^2 y}{d x^2}+ x\frac{d y}{d x}+ +(x^2-m^2)y =ay $ where $m$ is an integer. I have tried to complete the square which could be adjusted by complete the square. However, the result was no longer Bessel function of first kind(i.e. $J_n,Y_n$), as it become rational.
How to solve Bessel function with rational part?
Just take the $a$ to the other side. You get a Bessel equation of non-integer order $\sqrt{m^2+a}$, and its solutions are Bessel functions of that order.
$$ c_1 J_{\sqrt{m^2+a}}(x) + c_2 Y_{\sqrt{m^2+a}}(x) $$