 Laplace transform of the Bessel function J0

For real $p>0$, the Laplace transform $G$ of the order-zero Bessel function of the first kind satisfies $(1+p^2)G^{\prime }+pG=0$. This follows by transforming $x{J}_0^{\prime \prime }+J_0^{\prime }+x{J}_0=0$, using $\mathcal{L}[xg]=-{\partial }_p\mathcal{L}[g]$ and $J_0(0)=1$, $J_0^{\prime }(0)=0$. Thus $G=C/\sqrt{1+p^2}$; the initial value theorem for Laplace transforms gives $pG(p)\to 1$ and fixes $C=1$.