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Question-66827




Question Number 66827 by mr W last updated on 20/Aug/19
Commented by mr W last updated on 20/Aug/19
the graph above shows the equation  ((ln (x+(√(1+x^2 ))))/( (√(1+x^2 ))))=((ln (y+(√(1+y^2 ))))/( (√(1+y^2 ))))    it contains in fact two curves:  curve 1: y=x which always fulfills  the equation  curve 2: y=???    can you find the function for curve 2  in form of y=f(x)?  can you find the intersection point  of both curves? i mean with exact  values?
$${the}\:{graph}\:{above}\:{shows}\:{the}\:{equation} \\ $$$$\frac{\mathrm{ln}\:\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\right)}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}=\frac{\mathrm{ln}\:\left({y}+\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }\right)}{\:\sqrt{\mathrm{1}+{y}^{\mathrm{2}} }} \\ $$$$ \\ $$$${it}\:{contains}\:{in}\:{fact}\:{two}\:{curves}: \\ $$$${curve}\:\mathrm{1}:\:{y}={x}\:{which}\:{always}\:{fulfills} \\ $$$${the}\:{equation} \\ $$$${curve}\:\mathrm{2}:\:{y}=??? \\ $$$$ \\ $$$${can}\:{you}\:{find}\:{the}\:{function}\:{for}\:{curve}\:\mathrm{2} \\ $$$${in}\:{form}\:{of}\:{y}={f}\left({x}\right)? \\ $$$${can}\:{you}\:{find}\:{the}\:{intersection}\:{point} \\ $$$${of}\:{both}\:{curves}?\:{i}\:{mean}\:{with}\:{exact} \\ $$$${values}? \\ $$
Commented by Prithwish sen last updated on 23/Aug/19
Sir ,is there any process of decomposing  such functions ?
$$\mathrm{Sir}\:,\mathrm{is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{process}\:\mathrm{of}\:\mathrm{decomposing} \\ $$$$\mathrm{such}\:\mathrm{functions}\:? \\ $$
Commented by mr W last updated on 24/Aug/19
I don′t know, sir.  That′s also my question.  A famous example is x^y =y^x .
$${I}\:{don}'{t}\:{know},\:{sir}. \\ $$$${That}'{s}\:{also}\:{my}\:{question}. \\ $$$${A}\:{famous}\:{example}\:{is}\:{x}^{{y}} ={y}^{{x}} . \\ $$

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