Partial integrals of ordinary differential systems

TL;DR

This paper studies various types of partial integrals in ordinary differential systems, exploring their properties, construction methods, and applications to classical problems like Darboux's, including solving inverse problems.

Contribution

It introduces new methods for constructing first integrals and last multipliers from partial integrals and solves the inverse problem for differential systems based on their partial integrals.

Findings

Constructed integral basis for the Jacobi system

Developed methods for building first integrals from partial integrals

Solved the inverse problem of differential systems based on partial integrals

Abstract

Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last multipliers by known partial integrals are considered. Applications of partial integrals to solve the Darboux problem and to the extended Darboux problem are given. Integral basis of the Jacobi system is built. And the inverse problem of constructing differential systems on the base of their partial integrals is solved.