Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2005 / Paper 2 / 5 / Solution

The one-sided Laplace transform is $\bar{u}(p)={\int }_0^{\infty }{e}^{-pt}u(t)dt$ for sufficiently large $\mathrm{Re}p$. Integration by parts, with the exponential suppressing the upper boundary, givesDifferentiation under the integral gives $d\bar{u}/dp=-{\int }_0^{\infty }t{e}^{-pt}u(t)dt$, proving $\overline{tu}=-d\bar{u}/dp$. Start with $\overline{e^{-t}}=1/(p+1)$ and apply this rule repeatedly:With all three initial data zero, transforming the forced equation gives $(p+1{)}^3\bar{y}=1/(p+1)$. Thus $\bar{y}=(p+1{)}^{-4}$ and the listed inverse transform givesThe cubic factor is also the resonant exponential particular solution associated with the triple characteristic root $-1$.