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Taking the Laplace transform, using linearity and the preceding three formulas, givesSince $x(0)=0$, division by $p-1$ gives $\bar{x}(p)=2/p^3$ for $p\ne 1$ in the domain of convergence. There is no genuine singularity at $p=1$: continuity of the Laplace transform there supplies the same value. One can also first work with $\mathrm{Re}p>1$, where division is harmless.

The formula in part (iii) identifies $2/p^3$ as the Laplace transform of $t^2$. By uniqueness of the Laplace transform,As a direct check, $\dot{x}-x=2t-t^2$ and $x(0)=0$. Any two solutions of this initial value problem differ by $C{e}^t$ and have the same initial value, forcing $C=0$. This also verifies uniqueness independently of transform inversion.

Put $G(p)=\mathcal{L}[J_0](p)$, initially for real $p>0$. Besides the Laplace transform of a derivative, use multiplication by the original variable in the formTransforming the Bessel differential equation, with its given initial conditions, yieldsThe constant terms cancel, leaving $(1+p^2)G^{\prime }+pG=0$. This first-order ordinary differential equation integrates toThe initial value theorem for Laplace transforms gives ${\lim }_{p\to \infty }pG(p)=J_0(0)=1$, so $C=1$. Thus the Laplace transform of the Bessel function J0 isFor complex $p$ with $\mathrm{Re}p>0$, choose the analytic square-root branch agreeing with the positive square root on the positive real axis.

For the forcing $k=J_0$, part (b) now gives $F(p)=(1+p^2{)}^{-3/2}$. On the other hand, the same multiplication rule givesTaking an integral from zero divides the Laplace transform by $p$. By uniqueness of the Laplace transform,A direct check also explains the cancellation. This expression has $f^{\prime }=x{J}_0$ and $f^{\prime \prime }=J_0+x{J}_0^{\prime }$. The Bessel differential equation says $(x{J}_0^{\prime }{)}^{\prime }=-x{J}_0$, so integration from zero gives $x{J}_0^{\prime }=-{\int }_0^{x}t{J}_0(t)dt=-f$. Hence $f^{\prime \prime }+f=J_0$, and both initial conditions hold.