[ First time I post here I hope this is the right community where to ask this. To those who 'evaluate' negatively the question please provide some tips on how to make it 'better', so far i've tried my best by showing explicitly what functions i face. Many thanks :) ]
In my problem I find myself with certain $ x^nf(x) $ forms. $f(x)$ is a well-known solution of a certain class of differential equations. I wonder if we can infer some 'originating' differential equation that would give rise to $x^nf(x)$ solutions? $n$ can be integer or half-integer.
For instance, I have functions involving Bessel solutions ($I$ or $Y$ or $K$...), so I have forms such as $K_0(x)$, $ \sqrt{x}K_{1/2}(x)$, $xK_1(x)$. I wonder if these can be traced back to an originating differential equation.
I have not managed to find resources on this, or maybe haven't formulated my question properly to get to the right resources...
Any insight would be appreciated!
Thank you
If the differential equation for the known function is not too complicated, this can sometimes be done naively by hand.
For example, consider the Bessel equation $$x^2 y'' + x y' + (x^2 - \alpha^2) y = 0 .$$
For a solution $y$, set $z := x y$, so that $$z' = x y' + y, \qquad z'' = x y '' + 2 y' .$$
This motivates rewriting the Bessel equation in terms of these expressions, by writing expressions in terms of $y$ in terms of $z$, starting with the highest-order terms First as $$x \underbrace{(x y'' + 2 y')}_{z''} - x y' + (x^2 - \alpha^2) y = 0 ,$$ and then as $$x z'' - \underbrace{(x y' + y)}_{z'} + (x^2 - \alpha^2 + 1) y = 0$$ and then as $$x z'' - z' + \left(x + \frac{1 - \alpha^2}{x}\right) \underbrace{(x y)}_z = 0,$$ giving $$\color{#bf0000}{\boxed{x z'' - z' + \left(x + \frac{1 - \alpha^2}{x}\right) z = 0}} .$$