Inverse problem for differential systems having a singularity and turning point of even or odd order

Abstract

In this paper‎, ‎the canonical property of the solutions and the inverse problem for a system of differential equations having a singularity and turning point of even or odd order are investigated‎. ‎First‎, ‎we study the infinite product representation of the solutions of the system in turning case‎, ‎and derive the corresponding dual equations‎. ‎Then‎, ‎by a replacement‎, ‎we transform the system of differential equations to a second-order differential equation with a singularity and find the canonical product representation of its solution‎, ‎and provide a procedure for constructing the solution of the inverse problem‎. We present a new approach to solve the inverse problems having a singularity inside the interval‎.