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  article.
Looking the equation. Free term.
Theorem-1: 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
Looking the involution is take notice at the equality (1), exchange the this,
following form equality
in accordance the equality (3) and (2)
The second order, vulgar differential equation integration problem is form
exchanged elementary as a result is property of the involution that equation (1) is solution of problem at the this equation.
2. Following, looking is equation
Theorem-2: If be that is involution at the equation (5), then equation (5) limit step after the problem be solved is form fourth order the equation differential
Prove: We have (6) at the equality (5)
The equality (6) differential ling sequence twice:
Above the equality is division
equality be used at the property of the involution, exchanged we have following equality
The and equates putting off equality (7)
Therefore, following bring in put.
So, make up the result of upper result is simplify s following aspect
Equation (8) we know coming the fourth order, constant coefficient equation differential
3. Following, looking equation
Teorema-3: If be that is involution at the equation (9), then equation (9) limit step after the problem be solved is form sixth order the equation differential.
Prove: We has (10) at the equation (9).
The equality (10) time three is differentia ling sequence:
The property of involution issuing exchange at the
Following equality is pass (11)
Above we are find , and at the (10) equality, so, that putting equality (11).
As a result:
The equality constant coefficient sixth order, equality of differential.
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