 Solution

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.