Differential Invariants for quasi-linear and semi-linear wave-type equations

In this paper, we consider the class of wave-type equations $u_{tt}=f(x,t,u)u_{xx}+g(x,t,u,u_x,u_t)$ and two special cases of it. We derive the equivalence transformations for these equations and using these transformations, we obtain differential invariants and invariant equations. We employ these invariants or/and invariant equations to classify all equations of this general class that can be mapped into a simple linear hyperbolic or into an elliptic equation. Additional applications are presented.