Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 9 / b / i

The characteristic equations for a quasilinear equation here areSuccessive integration gives the general characteristicwhere $a,b,c$ are integration constants. Equivalently $z-2x$ and $y-z^2/4$ are constant along each curve: differentiating them with respect to $s$ gives zero. Three constants occur because shifting the origin of the parameter $s$ describes the same geometric curve again; there are two independent curve invariants.