An equation of the curve satisfying xdy - ydx - x 2- y 2 dx and y 1-0 isA- y-2 x2x-1B- y-x2log-sin xC- y - x sin log - x -D- y2-xx-12

The correct option is C y = x sin (log|x|)The equation can be written as x^2 xdy ydx/x^2 = x √(1 (y/x)^2)dx ⇒d(y/x)/√(1 (y/x)^2) = dx/x⇒ sin^ 1y/x = log ...

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Standard XII

Mathematics

Variable Separable Method

An equation o...

Question

An equation of the curve satisfying $x d y - y d x = \sqrt{x^{2} - y^{2}} d x$ and y(1) = 0 is

y = x^{2} l o g | s i n x |

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y = x s i n (l o g | x |)

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y^{2} = x (x - 1)^{2}

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y = 2 x^{2} (x - 1)

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Solution

The correct option is B $y = x s i n (l o g | x |)$
The equation can be written as $x^{2} \frac{x d y - y d x}{x^{2}} = x \sqrt{1 - (\frac{y}{x})^{2}} d x$
$\Rightarrow \frac{d (\frac{y}{x})}{\sqrt{1 - (\frac{y}{x})^{2}}} = \frac{d x}{x} \Rightarrow s i n^{- 1} \frac{y}{x} = l o g | x | +$ const
Since y(1) = 0 so const =0. Hence y = x~ sin(log|x|)

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An equation of the curve satisfying xdy−ydx=√x2−y2dx and y(1) = 0 is

The correct option is B y=x sin(log|x|)The equation can be written as x2xdy−ydxx2=x√1−(yx)2dx ⇒d(yx)√1−(yx)2=dxx⇒sin−1yx=log|x|+const Since y(1) = 0 so const =0. Hence y = x~ sin(log|x|)

An equation of the curve satisfying $x d y - y d x = \sqrt{x^{2} - y^{2}} d x$ and y(1) = 0 is