For the differential equation, general solution for x , ( y x ) (… - Vidyadip

MCQ

For the differential equation, general solution for $x\,\cos \left( {\frac{y}{x}} \right)\left( {ydx + xdy} \right) = y\,\sin \left( {\frac{y}{x}} \right)\left( {xdy - ydx} \right)$ , (where $c$ is constant of integration) is

A
$x = cy\,\sec \left( {\frac{y}{x}} \right)$
✓
$xy\,\cos \left( {\frac{y}{x}} \right) = c$
C
$x = cy\,\sec \left( {\frac{x}{y}} \right)$
D
$xy = c \cdot \cos \left( {\frac{y}{x}} \right)$

✓

Answer

Correct option: B.

$xy\,\cos \left( {\frac{y}{x}} \right) = c$

b
$ y d x+x d y= \frac{y}{x} \cdot \tan \left(\frac{y}{x}\right)(x d y-y d x) $

$=x y \tan \left(\frac{y}{x}\right)\left(\frac{x d y-y d x}{x^{2}}\right) $

$ \frac{y d x+x d y}{x y} =\tan \left(\frac{y}{x}\right) d\left(\frac{y}{x}\right) $

$ \frac{d(x y)}{x y} =\tan \left(\frac{y}{x}\right) \cdot d\left(\frac{y}{x}\right) $

$ \operatorname{In}(x y) =\ln \left(\sec \left(\frac{y}{x}\right)\right)+\ln c $

$ x y =c \cdot \sec \left(\frac{y}{x}\right) $

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