It is certainly straightforward to numerically solve the two ordinary differential equations of the previous subsection along a characteristic line using say a Runge-Kutta method. You would need to start from some point at which an initial or boundary condition is given.
If you find the solution along each of a densely spaced set of characteristic lines, you have essentially found everywhere.
Of course, if is zero somewhere in the region of interest, it may
be a better idea to find and as functions of instead of
and as functions of , by taking suitable ratios from
(3.4). Or you could just find all three variables as
function of the arc length along the characteristic lines, by
solving