Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 10 / c

The operator is $(D+1{)}^2$. Set $y=e^{-x}u$; direct differentiation gives $(D+1{)}^{2}y=e^{-x}{u}^{\prime \prime }$. Hence $u^{\prime \prime }=3x^2$, and two integrations giveThe two free constants multiply the independent homogeneous solutions $e^{-x}$ and $x{e}^{-x}$.