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

The integrating factor is $e^{2x}$, giving $(e^{2x}y{)}^{\prime }=1+e^{2x}(x^2+2x)$. A quadratic particular solution for the polynomial forcing is $x^2/2+x/2-1/4$, while the resonant exponential forcing contributes $x{e}^{-2x}$. ConsequentlySubstitution gives $y^{\prime }+2y=e^{-2x}+x^2+2x$, and the free multiple of $e^{-2x}$ is the homogeneous solution.