I have this second-order differential equation:
$$x''(t) + \frac{1}{(\tau + t)}x'(t) + k^2x(t) = 0$$
I want to make the solution to this ODE amenable to a closed form Bessel function, and so a suggested way is to make a change of variables so that we can compare the differential equation above to the transformation equation below: (where this $x$ is analogous to my $t$, and this $y$ is analogous to my $x(t)$)
Transformation equation:
The goal (or atleast the way I did it for a simple function) was to compare and identify what values the parameters $\alpha, \beta, C, m$ must have so that the form of differential equation is captured by a Bessel function that makes use of these parameters (such as a linear combination of $x^{\alpha}J_m(Cx^{\beta})$ and $x^{\alpha}Y_m(Cx^{\beta})$). This method allowed me to solve a simple equation like the Airy equation. But if I try to do that in this case, the moment I divide the boxed equation on both sides by $x^2$, you get a $\frac{1}{x}$ as the co-efficient for the first-derivative term, which doesn't represent the form of my differential equation's 2nd term (which has $\frac{1}{\tau + t}$ as its coefficient).
I am wondering if I am missing something here, or perhaps there's an intermediary step that's required before I can use this method. Ultimately, I just need a solution to that differential equation that is represented as a Bessel function.
Hint
Changing variable $u=k(\tau+t)$ would lead to a very simple Bessel equation in $x(u)$