Formal solutions and a related equation for a class of fourth order differential equations of a hydrodynamic type

over a z-domain that includes a turning point, and the construction of a related equation. Elucidations of this statement are the following: (i) The differential equation (1.1) is one in which X is a parameter, real or complex, whose absolute value is large, namely j X I > N with some suitably large positive constant N. For generality, X will be dealt with as complex. (ii) The variable z ranges over a closed bounded domain, in which certain hypotheses, that will be enunciated, are fulfilled. It may be real or complex. For generality it will be dealt with as complex. (iii) The coefficient functions P(z, X), Q(z, X), and R(z, X) are power series in 1/X, namely