# 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??

30 Points
13 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.

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