Ideals of Differential Operators and Transformations of Linear Partial Differential Equations

摘要

A ring of linear differential operators with smooth coefficients generated by two differentiations is considered. Concepts of operators closed with respect to commutation, a resultant of two operators, and a two-dimensional analogue of Wronskian are introduced. Sufficient conditions that two differential operators are generators of a left ideal annihilating a finite-dimensional space of functions are found. Differential operators annihilating given functions are constructed. The operators obtained transform solutions of one second-order differential equation into solutions of another equation of the same order.