A) \[\frac{1}{xy}+\log |y|=c\]
B) \[-\frac{1}{xy}+\log |y|=c\]
C) \[\frac{1}{xy}+2\log |y|=c\]
D) \[\log |y|=cx\]
Correct Answer: B
Solution :
[b]: The given differential equation is \[ydx+(x+{{x}^{2}}y)dy=0\] \[\Rightarrow \]\[ydx+xdy+{{x}^{2}}ydy=0\Rightarrow d(xy)=-{{x}^{2}}ydy\] \[\Rightarrow \]\[\frac{d(xy)}{{{(xy)}^{2}}}=-\frac{dy}{y}\Rightarrow \int_{{}}^{{}}{\frac{d(xy)}{{{(xy)}^{2}}}}=-\int_{{}}^{{}}{\frac{dy}{y}}\]\[\Rightarrow \]\[-\frac{1}{xy}=-{{\log }_{e}}|y|+c\], which is the required solution of the given differential equation.
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