Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 5 / Solution

Rearrange as $y^{\prime }+(x-1)y=x-1$ and put $w=y-1$. The resulting linear differential equation is $w^{\prime }+(x-1)w=0$, soThe initial condition gives $C=1$. Its graph is a Gaussian function above the horizontal asymptote $y=1$, symmetric about $x=1$, with its maximum $y=2$ there. It rises for $x<1$ and falls for $x>1$, since $y^{\prime }=-(x-1)e^{-(x-1{)}^2/2}$.