First slide

Methods of solving first order first degree differential equations

Question

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

Moderate

A

$y = x^{2} \log |\sin x|$
B

$y = x \sin (\log |x|)$
C

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

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

Solution

$The equation can be written as x^{2} \frac{x d y - y d x}{x^{2}} = x \sqrt{1 - (y / x)^{2}} d x$
$\Rightarrow \frac{d (y / x)}{\sqrt{1 - {(y / x)}^{2}}} = \frac{d x}{x} \Rightarrow \sin^{- 1} y / x = \log |x| + c o n t$
$Since y (1) = 0 so const = 0 . Hence y = x \sin (\log | x |)$

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