A) \[{{\tan }^{-1}}\left( \frac{x}{y} \right)+\log y+c=0\]
B) \[2{{\tan }^{-1}}\left( \frac{x}{y} \right)+\log x+c=0\]
C) \[\log (y+\sqrt{{{x}^{2}}+{{y}^{2}}})+\log y+c=0\]
D) \[{{\sin }^{-1}}\left( \frac{x}{y} \right)+\log y+c=0\]
Correct Answer: A
Solution :
Given that,\[{{y}^{2}}dx+({{x}^{2}}-xy+{{y}^{2}})dy=0\] \[\Rightarrow \] \[dx+\frac{{{x}^{2}}-xy+{{y}^{2}}}{{{y}^{2}}}dy=0\] \[\Rightarrow \] \[\frac{dx}{dy}+{{\left( \frac{x}{y} \right)}^{2}}-\left( \frac{x}{y} \right)+1=0\] Let \[v=\frac{x}{y}\Rightarrow x=vy\] \[\Rightarrow \] \[\frac{dx}{dy}=v+y\frac{dv}{dy}\] \[\therefore \] \[v+y\frac{dv}{dy}+{{v}^{2}}-v+1=0\] \[\Rightarrow \] \[y\frac{dy}{dx}=-({{v}^{2}}+1)\] \[\Rightarrow \] \[\frac{dv}{{{v}^{2}}+1}+\frac{dy}{y}=0\] On integrating, we get \[{{\tan }^{-1}}v+\log y+c=0\] \[\Rightarrow \] \[{{\tan }^{-1}}\frac{x}{y}+\log y+c=0\]
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