A) \[-\frac{8}{9}{{\pi }^{2}}\]
B) \[-\frac{4}{9}{{\pi }^{2}}\]
C) \[-\frac{4}{9\sqrt{3}}{{\pi }^{2}}\]
D) \[\frac{-8}{9\sqrt{3}}{{\pi }^{2}}\]
Correct Answer: A
Solution :
\[\frac{dy}{dx}+y\cot x=\frac{4x}{\sin x}\] Integrating factor \[\text{=}{{\text{e}}^{\int_{{}}^{{}}{\cot x\,\,dx}}}\] \[={{e}^{+\,\,In\,\,|\sin x|}}\] \[=\sin x\] \[(\because x\in (0,\pi ))\] \[\therefore y\cdot (\sin x)=\int_{{}}^{{}}{\frac{4x}{\operatorname{sinx}}(\sin x)\cdot dx}\] \[y\cdot (\sin x)=2{{x}^{2}}+c\] \[y\left( \frac{\pi }{2} \right)=0\Rightarrow c=-\frac{{{\pi }^{2}}}{2}\] \[y\left( \frac{\pi }{6} \right)\] \[\therefore y.\left( \sin \frac{\pi }{6} \right)=2{{\left( \frac{\pi }{6} \right)}^{2}}-\frac{{{\pi }^{2}}}{2}\] \[y=\frac{-8}{9}{{\pi }^{2}}\]
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