So the question is trying to transform in the hyperbolic case, the PDE
$$a(x, y)u_{xx} + 2b(x, y)u_{xy} + c(x, y)u_{yy} + F(x, y,u,u_x,u_y)= 0$$
can be transformed to the normal form
$$\widetilde{u}_{ξ\eta} + {\Psi}(ξ, \eta , \widetilde{u}, \widetilde{u}_ξ,\widetilde{u}_η)$$
I know the steps I need to follow for this. However I don't know how to apply certain ones. For instance what does F represent in this case. Where would it fall in the general second order PDE. For example, here A = a, B = 2b, and C = 1. Am I on the right track? I just need to know what each constant is and how to deal with F and I can move forward from there.
Thanks for the help.