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if y=x+1/x+1/x+1/x+............ prove that (x^2-y^2+3)dy/dx=1 note that the + sign in the sum goes only with the denominator and not with the only sum so, we can sum it as y=x+1/y then what should be done??

if y=x+1/x+1/x+1/x+............
prove that
(x^2-y^2+3)dy/dx=1
note that the + sign in the sum goes only with the denominator and not with the only sum so,
we can
sum it as y=x+1/y
then what should be done??

Grade:10

1 Answers

Priyansh Bajaj AskiitiansExpert-IITD
30 Points
10 years ago

Dear Aritra,

Solution:- As you said, we can write the given eq. y = x + 1/(x+ 1/(x+ 1/(x+...))) as y = x+ 1/y. Then, you should differentiate the implicit eq. and evaluate dy/dx. Then, do some manipulation to achieve desired results. You can refer the following solution-

              y = x +1/y.................(1)

              dy/dx = 1 - (1/y2) (dy/dx)

           [1 + (1/y2)] (dy/dx) = 1..................(2)

        From eq. (1), y-1/y = x. On squaring both sides, we get-

 (y-1/y )2= x2 or  y2 + 1/y2 -2= x2 or 1/y2 = (x2 - y2+2)

Putting the value of 1/yin eq. (2), we get-

  [1 + (x2 - y2+2)] (dy/dx) = 1

(x2 - y2+3) (dy/dx) = 1 

Hence proved.                        

Please feel free to post as many doubts on our discussion forum as you can. If you find any question
Difficult to understand - post it here and we will get you the answer and detailed solution very quickly. We are all IITians and here to help you in your IIT JEE preparation.

All the best Aritra!!!

Regards,
Askiitians Experts
Priyansh Bajaj

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