The first-order differential isomorphisms of canonical hyperbolic equations

Background. To determine unknown solutions of canonical hyperbolic differential equations for functions of two variables, it seemed relevant to establish the connection of the first-order differential isomorphisms of these equations with Laplace transformations. Materials and methods. To study the first-order isomorphisms, the theorem on the representation of isomorphisms by linear differential translators is applied. Direct actions with differential operators are used. Results and conclusions. The theorem is proved that any first-order differential isomorphism between canonical differential equations with real analytic coefficients is a composition of the first-order and zero-order Laplace transformations. This makes it possible to expand the scope of application of classical Laplace transformations.