Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 2 / 13 / iii / Solution

The repeated characteristic root is $-1$, matching the exponential forcing. Put $y=e^{-x}w$; then $y^{\prime \prime }+2y^{\prime }+y=e^{-x}{w}^{\prime \prime }$, so $w^{\prime \prime }=8$. The zero initial conditions give $w(0)=w^{\prime }(0)=0$, henceThe extra quadratic factor is the resonance associated with the double root.