solve-for-x-y-R-1-x-2-ln-x-1-x-2-1-y-2-ln-y-1-y-2-

Question Number 66619 by mr W last updated on 17/Aug/19

s o l v e f o r x, y \in R

\frac{\sqrt{1 + x^{2}}}{\ln (x + \sqrt{1 + x^{2}})} = \frac{\sqrt{1 + y^{2}}}{\ln (y + \sqrt{1 + y^{2}})}

Answered by Smail last updated on 17/Aug/19

\Leftrightarrow \frac{{({s i n h}^{- 1} (x))}^{'}}{{s i n h}^{- 1} (x)} = \frac{{({s i n h}^{- 1} (y))}^{'}}{{s i n h}^{- 1} (y)}

\Rightarrow l n ({s i n h}^{- 1} (x)) = l n ({s i n h}^{- 1} (y)) + c

\Leftrightarrow l n ({s i n h}^{- 1} (x)) = l n ({k s i n h}^{- 1} (y))

{s i n h}^{- 1} (x) = {k s i n h}^{- 1} (y)

y = s i n h (\frac{{s i n h}^{- 1} (x)}{k})

Commented by mr W last updated on 17/Aug/19

t h a n k y o u s i r!

Commented by mr W last updated on 18/Aug/19

b u t s h o u l d i t n o t b e :

\frac{\ln (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} = \frac{\ln (y + \sqrt{1 + y^{2}})}{\sqrt{1 + y^{2}}}

\Leftrightarrow \sinh^{- 1} (x) {(\sinh^{- 1} (x))}^{'} = \sinh^{- 1} (y) {(\sinh^{- 1} (y))}^{'}

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