Not so easy

Calculus Level 3

$y =\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots }}}}$

The relationship between $x$ and $y$ are as shown above.

Which of the following is true?

Clarification : The RHS represents a nested function .

Here, $x>0$ .

y=\int \dfrac{dx}{2y-1}

y=\int\dfrac{\log(x\cdot y) \, dx}{x^{y}}

None of these choices

y=\int\dfrac{e^{y\cdot x} \, dx}{y}

1 solution

Aman Sharma
Aug 29, 2014

Equation can be written as a cyclic polynomial as:- $y=\sqrt{x+y}$ Squaring both sides:- $y^{2}=x+y$ Rearanging:- $y^{2}-y=x$ Diffatantiating both sides wrt x:- $2y\frac{dy}{dx}-\frac{dy}{dx}=1$ Rearranging:- $\frac{dy}{dx}(2y-1)=1$ $dy=\frac{dx}{2y-1}$ Integrating both sides:- $y=\int{\frac{dx}{2y-1}}$ . .

Bingooo

Technically, none of the options are correct, since the indefinite integral evaluates to $F(x) + C$ .

See A doubt on integration for a similar paradox.

Calvin Lin Staff - 6 years, 9 months ago

Well i was thinking that,since i didn't evaluted the intigral so adding a constent of integration is not the case

Aman Sharma - 6 years, 9 months ago

LOL , looks like this question is a "non-intended" troll ;P

Nihar Mahajan - 5 years, 2 months ago

honestly, I would have never thought of this. Ever.

Jojo Ofthemojo - 6 years, 9 months ago

Because Your missing basic .

Satyam Singh - 6 years, 9 months ago

wonderfull

Karudaiyar Ganapathy - 6 years, 9 months ago

What the hell? I have answered the correct answer while it shows that none of these? C can be 0 also!

Agastya Chandrakant - 6 years, 4 months ago

U found ans to be (B) BT u gave ans as (none of these) how ???

Mayank sood - 5 years, 9 months ago

The function has a discontinuity at 0, with $y(0)=0$ . You should require $x>0$ .

Otto Bretscher - 5 years, 3 months ago

Not so easy

1 solution

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