Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2012 / 9

For the Laplace transform of a derivative, first apply integration by parts on a finite interval:Under the usual transform hypotheses, including sufficient regularity for integration by parts, existence of the derivative's Laplace transform, and $e^{-pT}x(T)\to 0$, passing to the limit givesThese conditions hold, for example, for a continuously differentiable $x$ with $x$ and $\dot{x}$ of exponential order, in a sufficiently far right half-plane. Existence of $\bar{x}(p)$ alone does not establish all of these derivative-transform hypotheses; the finite-interval identity displays exactly which boundary term must disappear. The polynomial solution in the last part satisfies them for every $\mathrm{Re}p>0$.

For $\mathrm{Re}p>0$, the complex exponential $e^{-pT}$ tends to zero, since its modulus is $e^{-T\mathrm{Re}p}$. Therefore direct integration givesThe condition on the real part also gives absolute convergence of this improper integral.

Write $I_n(p)={\int }_0^{\infty }{e}^{-pt}{t}^{n}dt$. For $\mathrm{Re}p>0$, exponential decay dominates every fixed power of $t$, so the improper integral converges absolutely and $e^{-pT}{T}^n\to 0$. For $n\ge 1$, integration by parts givesThe preceding part gives the base case $I_0(p)=1/p$. Repeated application, or mathematical induction, yields

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.