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The solution of $(y (1 + x^{- 1}) + \sin y) d x + (x + \log_{e} x + x \cos y) d y = 0$ is

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1+y−1.siny+x−1.logex=C

y+siny+xylogex=C

xy+ylogex+xsiny=C

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detailed solution

Correct option is C

ydx+yx⋅dx+sin⁡ydx+xdy+loge⁡xdy+xcos⁡ydy=0⇒(ydx+xdy)+y⋅dxx+loge⁡xdy+(sin⁡ydx+xcos⁡ydy)=0⇒d(xy)+dyloge⁡x+d(xsin⁡y)=0 integrating on both sidesxy+yloge⁡x+xsin⁡y=C

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The solution of y1+x−1+sinydx+x+logex+xcosydy=0 is

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The solution of $(y (1 + x^{- 1}) + \sin y) d x + (x + \log_{e} x + x \cos y) d y = 0$ is