 Solution

For $p$ in a half-plane of absolute convergence, interchange the integrals using Fubini's theorem:For positive real $p$, integration by parts gives $I_n={\int }_0^{\infty }{e}^{-pt}{t}^{n}dt=n{I}_{n-1}/p$, starting from $I_0=1/p$. Hence $I_n=n!/p^{n+1}$, orThe formula extends throughout $\mathrm{Re}p>0$ by the same integration argument. Write $X=\mathcal{L}x$. The integral term is the convolution of $t$ with $x(t)$, whose Laplace transform is $X/p^2$. The derivative transforms to $pX-x(0)=pX-1$. Thereforeand elementary factorization givesInverting the two polynomial transforms yieldsDirect substitution verifies $x^{\prime }=t$ and ${\int }_0^t(t-s)(1+s^2/2)ds=t^2/2+t^4/24$, as required. Uniqueness also follows without transform assumptions: differentiating the equation twice gives $x^{(3)}+x=1+t^2/2$ with initial data $x(0)=1$, $x^{\prime }(0)=0$, $x^{\prime \prime }(0)=1$, a linear initial-value problem with a unique solution.