I'm looking at solving the Bessel Differential Equation of order $0$ using Laplace transforms. Starting from the equation
$$ tx'' + x' + tx = 0 $$
I can take the Laplace transform of the equation just fine to get the equation
$$ (s^2+1)X' + sX = 0 $$
and solving this gives me
$$ X(s) = c\frac{1}{\sqrt{s^2+1}} $$
I can show via power series argument that the inverse Laplace transform gives me
$$ x(t) = cJ_0(t) $$
which of course is a solution to the equation. But what happened to my other solution? I know that this equation should have solution
$$ x(t) = cJ_0(t) + k Y_0(t) $$
so where did half of my solution go? My first guess was that maybe $Y_0$ doesn't have a Laplace transform, but it turns out that
$$ \mathcal{L}\{-\frac{\pi}{2}Y_0(t)\} = \frac{\sinh^{-1}s}{\sqrt{s^2+1}} $$
which led me to try and reverse engineer which first order ODE would give me this as a solution, when I discovered that
$$ (s^2+1)X' + sX = \kappa $$
would in fact give me
$$ X(s) = c\frac{1}{\sqrt{s^2+1}} + \kappa\frac{\sinh^{-1}s}{\sqrt{s^2+1}} $$
which would invert to the proper solution (with $k = -\frac{\pi}{2}\kappa$). So the question becomes, what happened to my $\kappa$ constant? I checked and double checked my math by hand and using Wolfram|Alpha; differentiation kills $x'(0)$, and that plus the minus sign cancels out $x(0)$. How can I justify Laplace transform killing one of my solutions by dropping a constant? And can I fix issues like this in general just by adding in my own constant like I could here?
Edit: I just want to add that I know that using the Laplace transform on the original equation
$$ t^2x'' + tx' + t^2x = 0 $$
would preserve the solution, but (1) the Laplace transform doesn't simplify this version of the equation quite as much, and (2) it shouldn't matter because both equations have the same set of solutions.