Инволюция, уравнение Зильберштейна, обыкновенные дифференциальные уравнения с инволюцией, неоднородные дифференциальные уравнения с инволюцией.
In this paper is considered the problem to solve the second order no ordinary differential equation with involution.
Definition: If do reflected
, equality
. This reflect is involution.
Such functions 
arrange an involution.
The properties of involution is take possession a lot of problem full give at the [1] article.
Following
(1)
Looking the equation.
Free term.
Theorem: If be 
that is involution 
at the equation (1), then equation (1) limit step after the problem be solved is form second order the equation differential.
Prove: The equality (1) for be solved, this the is equality the aspect writing
its differentiated, as a result is form equality
(2)
Looking the involution is take notice at the equality (1), exchange the 
this
,
following form equality
(3)
in accordance the equality (3) and (2)
The second order, vulgar differential equation integration problem is form
(4)
exchanged elementary as a result is property of the involution that equation (1) is solution of problem at the this equation. 
Example:
Accordance (4) equality solution
(5)
Mean
-
If be (1) equality being
(6)
(4) equality as being
(7)
or
Also,
(8)
Thisequality
,
,
Reference:
- Винер И. Я. Дифференциальные уравнения с инволюциями. // Дифференциальные уравнения. Том 5, 1969.
- Хромов А. П. Смешанная задача для дифференциального уравнения с инволюцией и потенциалом специального вида. // Известия Саратовского Университета, Нов.сер. Математика, Механика, Информатика. 2010,,т.10. вып № 4,с.17–22.
- Курдюмов В. П., Хромов А. П. О базисах Рисса из собственных и присоединенных функций функционально-дифференциального оператора переменной структуры. //Изв. Сарат.унта. Нов.сер.Математика. Механика. Информатика.2007.Т.7, вып.2, С. 20–25.
