Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On Explicit Integration of Differential Inequalities

Author(s):

Yu. A. Il'in

Ph.D. in Math.Sci.,
St.Petersburg State University,
Faculty of Mathematics and Mechanics

iljin_y_a@mail.ru

Abstract:

In the article we consider the problem of finding in explicit form all the solutions of a first order differential inequality which is obtained from a differential equation integrated in quadratures. In contrast to the comparison theorems method (Chaplygin method), we are interested not in obtaining estimations on solutions, but in derivation a formula describing all the functions which satisfy the given inequality. The main technique is the change of variable in inequality by using the formula of the general solution of the corresponding equation (method of variation of arbitrary constant). We discuss in detail the difficulties encountered, including the problem of the maximum extension of the inequality solutions. In the paper we consider both a general case, and the examples of specific inequalities obtained from differential equations of classical integrable types.

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