I have a highly nonlinear PDE, in a form
D[f[x,y],x]^2==G
where
f[x,y] is the function I wanna solve. G is some function of f[x,y], D[f[x,y],x], x and y.
So,there are two solutions:D[f[x,y],x]=-Sqrt[G],and D[f[x,y],x]=+Sqrt[G]
after specify some initial conditions, I can first solve the first one (- sign), numerically. then at some point, this solution terminates because of singularies or something else (the solution, which is a interpolating function, will turn from a real value to a complex value). The terminating point is a line y0=y0. At this point I want to change the equation to the (+ sign) one, with the new initial conditions given by f[x,y0[x]].
How can I do the above steps?
Especially, first, how can I get this terminating line y0=y0; second, how can I specify f[x,y0] as the new initial conditions?
By the way, how to make Mathematica solve PDE only in real number field?