As part of my research I get the following differential equation. I need to solve for <mrow class

Ayaan Barr 2022-07-09 Answered
As part of my research I get the following differential equation. I need to solve for V ( γ ). In fact the requirement is not to solve but to show that V ( γ ) is monotonic in a j j, (which I hope it is) where a j are positive valued constants which do not depend on γ. If it can be shown without solving the differential equation that is sufficient. Please provide some suggestions.
γ log ( e ) d d γ V ( γ ) = 1 η ( γ )
η ( γ ) = 1 1 + γ j a j 1 a j ( γ η ( γ ) )
where j = { 1 , , n }
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Answers (1)

Tristin Case
Answered 2022-07-10 Author has 15 answers
It seems that η a ( γ ) solves the equation F γ ( η a ( γ ) , a ) + n = 1 where
F γ ( h , a ) = h i 1 1 + γ h a i .
Each function F γ (   , a ) is increasing. Each function F γ ( h ,   ) is increasing. Hence a η a ( γ ) is decreasing, for each γ. This proves that a d d γ V a ( γ ) is increasing.
Thus, assuming that there exists some γ 0 such that V a ( γ 0 ) does not depend on a, one sees that a V a ( γ ) is increasing if γ > γ 0 and decreasing if γ < γ 0 .
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