We consider a quasilinear hyperbolic homogeneous system of two first order equations involving two dependent and two independent variables. For the associated hodograph equations we investigate the reducibility tio canonical forms allowing for an explicit integration. Such a kind of requirement, in problems of physical interest, provides suitable method for characterizing possible model alws. The theoretical approach shown herein can be relevant for studying the existence of conservation laws to non-homogeneous first order systems and also for describing the evolution of weak shock waves.
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