If y- x sin-1x-1-x2- prove that 1-x2dydx - 1-xy

Consider y=x sin^ 1 x√(1 x^2) We have from Quotient Rule, ddx[uv]=vdudx udvdxv^2 Applying this we get dydx=1√(1 x^2).√(1 x^2) (sin^ 1 x).12(1 x^2)^ ...

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Byju's Answer

Standard XII

Mathematics

Variable Separable Method

If y= x sin...

Question

If $y = \frac{x s i n^{- 1} x}{\sqrt{1 - x^{2}}}$ , prove that $(1 - x^{2}) \frac{d y}{d x} = 1 + x y$

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Solution

Consider $y = \frac{x {sin}^{- 1} x}{\sqrt{1 - x^{2}}}$

We have from Quotient Rule,

$\frac{d}{d x} [\frac{u}{v}] = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}$

Applying this we get

$\frac{d y}{d x} = \frac{\frac{1}{\sqrt{1 - x^{2}}} . \sqrt{1 - x^{2}} - ({sin}^{- 1} x) . \frac{1}{2} (1 - x^{2})^{\frac{- 1}{2}} (- 2 x)}{1 - x^{2}}$

$\frac{d y}{d x} = \frac{1 + x \frac{{sin}^{- 1} x}{\sqrt{1 - x^{2}}}}{1 - x^{2}}$

$d y (1 - x^{2}) = (1 + x \frac{{sin}^{- 1} x}{\sqrt{1 - x^{2}}}) d x$

$d y (1 - x^{2}) = (1 + x y) d x$ ----------- Since, $y = \frac{x {sin}^{- 1} x}{\sqrt{1 - x^{2}}}$ --given

$(1 - x^{2}) \frac{d y}{d x} = (1 + x y)$

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Similar questions

Q. If

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

, prove that

(1 - x^{2}) \frac{d y}{d x} - x y = 1

State the whether given statement is true or false

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

, prove that

(1 - x^{2}) \frac{d y}{d x} = x + \frac{y}{x}

Q. If

y = \frac{x {sin}^{- 1} x}{\sqrt{1 - x^{2}}} + log \sqrt{1 - x^{2}}

then prove that

\frac{d y}{d x} = \frac{x \sqrt{1 - x^{2}} + {sin}^{- 1} x (1 + x^{2}) - 2 x (1 - x^{2})}{\sqrt{1 - x^{2}} (1 - x^{2})}

Q. If

y = x \sin^{- 1} x + \sqrt{1 - x^{2}}

, prove that

\frac{d y}{d x} = \sin^{- 1} x

Q. If

y = \frac{\sqrt{1 - x}}{\sqrt{1 + x}}

, prove that

(1 - x^{2}) \frac{d y}{d x} + y = 0

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If y=xsin−1x√1−x2, prove that (1−x2)dydx=1+xy

If $y = \frac{x s i n^{- 1} x}{\sqrt{1 - x^{2}}}$ , prove that $(1 - x^{2}) \frac{d y}{d x} = 1 + x y$