Past exam of the differential equations 1 course of the University of Oxford / 2017 / 3 / c / Solution

On $y=1$, the transformed variables satisfy $s=1/3$ and $1/2\le r\le 1$. Differentiating the displacement data tangentially and using $u_y=1$ givesHence $g^{\prime }=2$ on $[5/6,4/3]$, $h^{\prime }=0$ on $[1/6,2/3]$, and the additive constants have sum $-2/3$. The uniquely determined solution is thereforeIt is fixed by the data exactly where both needed characteristic arguments belong to their known intervals:Equivalently, in the $(r,s)$ plane this is the diamond $|r-3/4|+|s-1/3|\le 1/4$. In the original coordinates it is bounded by four curved characteristic curves. It lies wholly in $x,y>0$. The displayed formula supplies one smooth extension beyond the diamond too, but the specified finite data segment does not determine that extension: one may change $g$ or $h$ outside its known argument interval. This is the characteristic-coordinate domain of dependence.