Past exam of the differential equations 1 course of the University of Oxford / 2021 / 2 / c / i / Solution

Treat both sides of $a$ at once: for $\sigma \in \{-1,1\}$ set $w(t)=v(a+\sigma t)$, $0\le t\le h$. Nonnegativity givesDefine $W(t)=ct+A{\int }_0^{t}w(r)dr$. By the fundamental theorem of calculus, $W^{\prime }(t)=c+Aw(t)\le c+AW(t)$ and $W(0)=0$. The integrating factor $e^{-At}$ givesSince $w\le W$, this proves the affine Gronwall estimate