Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2009 / 8 / Solution

The characteristic equations for a transport equation are $\dot{x}=1$, $\dot{y}=2$, $\dot{z}=-3z$. Label an initial point by $s\in [0,1]$ and choose characteristic time zero on the data segment. Integration givesThus $s=x-y/2$ and $t=y/2$, givingThis is the full infinite strip between the parallel characteristic lines $y=2x$ and $y=2x-2$. Both positive and negative characteristic times are allowed because the problem specifies no forward-time restriction. Each characteristic in this strip meets the data segment exactly once. Outside it, the general solution $z=e^{-3y/2}F(x-y/2)$ contains unrestricted values of $F$, so the data do not determine a unique solution there.