Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 15 / a / ii

The characteristic equation has a repeated root $-1$, suggesting the substitution $y=e^{-x}v$. By the product rule,so $y^{\prime \prime }+2y^{\prime }+y=e^{-x}{v}^{\prime \prime }$. The inhomogeneous linear differential equation reduces to $v^{\prime \prime }=2x$. Two integrations giveThereforeThe two arbitrary constants account for the repeated-root homogeneous solution.