Let a solution y=y(x) of the differential equation x x^2-1 d y-y y^2-1 d x=0 satisfy y(2)= 2 3 Statement 1 y(x)= ( ^ -1 x- (π)/(6)) Statement 2 y(x) is given by (1)/(y)= 2 3 x - 1-(1)/(x^2)

Correct answer: C. Statement 1 is true, Statement 2 is false. (d y)/(d x)= y y^2-1 x x^2-1 ∫ d y y y^2-1 =∫ d x x x^2-1 ^ -1 y= ^ -1 x+C At x=2, y= 2 3 (π)/(6)=(π)/(3)+ C C=- (π)/(6) Now, y= ( ^ -1 x-(π)/(6)) aligned & = [ ^ -1 (1)/(x)- ^ -1 3 2 ] & = [ ^ -1 ( 3 2 x + 1-(1)/(x^2) 1-(3)/(4) )] (1)/(y) & = 3 2 x +(1)/(2) 1-(1)/(x^2)

JEE Advanced 2008Paper 2

Exam: JEE Advanced
Year: 2008
Paper: Paper 2
Subject: Mathematics
Chapter: Differential Equations

JEE Advanced 2008 · Paper 2 · Mathematics · Differential Equations

Let a solution y=y(x) of the differential equation x x^2-1 d y-y...

Let a solution

y=y(x)

of the differential equation

x \sqrt{x^2-1} d y-y \sqrt{y^2-1} d x=0

satisfy

y(2)=\frac{2}{\sqrt{3}}

Statement

1 y(x)=\sec \left(\sec ^{-1}

\frac{\pi}{6}\right) Statement 2 y(x)

is given by

\frac{1}{y}=\frac{2 \sqrt{3}}{x}-\sqrt{1-\frac{1}{x^2}}

Options

AStatement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
BStatement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
CStatement 1 is true, Statement 2 is false.
DStatement 1 is false, Statement 2 is true

Correct answer

C. Statement 1 is true, Statement 2 is false.

Step-by-step solution

\because \frac{d y}{d x}=\frac{y \sqrt{y^2-1}}{x \sqrt{x^2-1}} \Rightarrow \int \frac{d y}{y \sqrt{y^2-1}}=\int \frac{d x}{x \sqrt{x^2-1}}\Rightarrow \quad \sec ^{-1} y=\sec ^{-1} x+C

\quad x=2, y=\frac{2}{\sqrt{3}}\frac{\pi}{6}=\frac{\pi}{3}+

C \Rightarrow C=-

\frac{\pi}{6}

Now,

y=\sec \left(\sec ^{-1} x-\frac{\pi}{6}\right) \begin{aligned} & =\cos \left[\cos ^{-1} \frac{1}{x}-\cos ^{-1} \frac{\sqrt{3}}{2}\right] \\ & =\cos \left[\cos ^{-1} \left(\frac{\sqrt{3}}{2 x}+\sqrt{1-\frac{1}{x^2}} \sqrt{1-\frac{3}{4}}\right)\right]

\\ \Rightarrow \quad

\frac{1}{y}

=\frac{\sqrt{3}}{2 x}+\frac{1}{2} \sqrt{1-\frac{1}{x^2}}

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