The introduction to the operator method for solving differential equations.First-order DE

Abstract

We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of the equations and then to transform them to more convenient form with help of developed family of operator identities. On example of non-linear first-order DEs we analyse analytical and algorithmical possibilities for solutions obtaining. Different forms of solutions for first-order DEs are given, including for some integro-differential equations and equations with variational derivatives. We describe new algorithms for direct computing the solutions with help of computer algebra system (CAS). We also discuss recipe for finding new solvability conditions, which allow to enlarge DE solving abilities of existent CAS.