The form
$$\Phi_s(p)= \int_0^\infty e^{-px} e^{-s/x} \, dx = 2\sqrt{\frac sp} K_1(2\sqrt{sp})$$
is a standard representation for the $K_\nu(\cdot)$ Bessel function ($\nu=1$). It appears in analytic number theory in some very deep, beautiful, and highly nontrivial ways.
I'm a little perplexed however that I've never seen anything in the literature or elsewhere that mentions that $\Phi_s(p)$ satisfies the following linear partial differential equation (resembling a linearized singular KdV equation):
$$ s^2 \frac{\partial^3 \Phi_s(p)}{\partial s^3} = p^2 \frac{\partial \Phi_s(p)}{\partial p} $$
This can be quickly proven using standard Bessel identities. Perhaps it's just not useful in analytic number theory? I don't know.
Question
Are there references which use this PDE in analytic number theory or related fields? Is the PDE just a re-packaging of the standard Bessel ODE somehow?