If the solution of differential equation $x d x - y d y = \frac{x}{\sqrt{x^{2} - y^{2}}} (x d y - y d x)$ is $y = y (x)$ and is passing through point $(1, 0) .$ Then the value of arbitrary constant involved is ?

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Solution

Given : $x d x - y d y = \frac{x}{\sqrt{x^{2} - y^{2}}} (x d y - y d x)$
put $x = r sec θ, y = r tan θ$
$\Rightarrow x^{2} - y^{2} = r^{2}, x d x - y d y = r d r, x d y - y d x = r^{2} sec θ d θ$
The equation becomes,
$r d r = \frac{r sec θ}{r} (r^{2} sec θ) d θ \Rightarrow \frac{d r}{r} = {sec}^{2} θ d θ$
integrating both sides we get,
$ln | r | = tan θ + c \Rightarrow \sqrt{x^{2} - y^{2}} = k e^{y / \sqrt{x^{2} - y^{2}}}$
where $k$ is arbitrary constant,
now at point $(1, 0) :$
$\Rightarrow k = 1$

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