On the asymptotic solutions of a class of ordinary differential equations of the fourth order, with special reference to an equation of hydrodynamics

The differentiations (indicated by accents or by a Roman superscript) are with respect to the (complex) variable z. For this variable a domain will be specified. The symbol X in the equation signifies a parameter. This is assumed to be large in absolute value and is dealt with also as being complex. The functions P(z, X), Q(z, X), and R(z, X), for such z and X are assumed to be expressible as power series in 1/X with coefficients that are functions of z thus