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

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.