Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2006 / Differential equations / 9

For a function for which the integral converges, the Laplace transform isFor a continuous function of exponential order this exists in a sufficiently far right half-plane. For the first input,This follows by evaluating the exponential antiderivative and using its decay at infinity.

Integration by parts, whose boundary term vanishes for $\mathrm{Re}p>0$, gives

Combining the exponential factors gives

The same integration by parts with $p$ replaced by $p+c$ givesFor the remaining unnumbered requests, the derivative and convolution formulas, under the usual convergence hypotheses, areApplying them to the general equation yields $p\bar{x}-a+\bar{k}\bar{x}=\bar{y}$ and thereforeFor the particular data, $\bar{y}=1/p$, $\bar{k}=1/(p+2)$ and $a=0$, soThe four transforms above giveAs a direct check, $x(0)=0$, $x^{\prime }=(1+t)e^{-t}$, andAdding the last two expressions gives one, verifying the original equation as well as the required limit.