Consider a vector field which can be written as system of separable differential equations, which I will solve below:
$$ X=\big\langle x\log x, -y \log y \big\rangle$$
for $x,y \in (0,1).$
I would like to transport this differential equation via Mellin transform. And so I've already solved the differential equation and mapped the integral curves into a different function space with the Mellin transform. The functions I obtain in the different function space are modified Bessel functions of the second kind, with one such integral curve taking the form:
$$f(s)=\frac{2K_1(2\sqrt{s})}{\sqrt{s}} $$
I'm aware of the modified Bessel differential equation of which the modified bessel function is a solution. But I'm not sure if this will be the exact form of the differential equation in the new function space or some different differential equation.
How can I finish the steps to obtain the transported differential equation?
I also tried using properties of the Mellin transform to change the differential equation into a different form.
Here is my solution to: $$ X=\big\langle x\log x, -y \log y \big\rangle$$
for $x,y \in (0,1).$
$\frac{dx}{dt} = x\ln(x)$
$\frac{dy}{dt} = -y\ln(y) \Rightarrow \frac{dx}{x\ln x} = \frac{dy}{-y\ln y} \Rightarrow \ln(\ln x) = -\ln (\ln y) + c \Rightarrow e^{\ln (\ln y)} = e^{-\ln(\ln x)+c} \\ \Rightarrow \ln y = \frac{s}{\ln x} \Rightarrow \ln x \ln y = s$
Re-arranging we get $y=f_s(x)=e^{\frac{s}{\log x}}$ and so we can map each of these integral curves to a different function space via the Mellin transform:
$$ f(s)=\frac{2K_1(2\sqrt{s})}{\sqrt{s}}=\int_0^1 x^{s-1} e^{\frac{1}{\log x}}~dx$$
Thus acting on each integral curve in the family successively, we obtain the entire family of integral curves in the new function space:
$$ f_t(s)=2 \sqrt{\frac{t}{s}}K_1(2\sqrt{t s}) $$
for $t>0.$
Much like $f_s(x)$ analytically foliates $M=(0,1)^2$ these "Bessel" integral curves analytically foliate the bounded region below $g(x)=1/x$ for positive $x,g.$
So I'm curious how to complete the argument and write down a vector field, related to $X$ by the Mellin transform, whose solutions are these Bessel integral curves.