Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 4 / b / i / Solution

For $x>0$, substitute $v=y/x$. Then $y^{\prime }=v+x{v}^{\prime }$, and cancellation reduces the differential equation to $v^{\prime }+2v=e^{-2x}$. Multiplication by the integrating factor $e^{2x}$ gives $(e^{2x}v{)}^{\prime }=1$. ThereforeDirect substitution gives $y^{\prime }+2y-y/x=x{e}^{-2x}$ and verifies the result. Dividing by $x$ is legitimate on the stated positive domain.