A) \[-\frac{{{\pi }^{2}}}{4\sqrt{3}}\]
B) \[-\frac{{{\pi }^{2}}}{2}\]
C) \[-\frac{{{\pi }^{2}}}{2\sqrt{3}}\]
D) \[\frac{{{\pi }^{2}}}{2\sqrt{3}}\]
Correct Answer: C
Solution :
\[\frac{dy}{dx}-y\tan x=6x\sec x\] \[y\left( \frac{\pi }{3} \right)=0;y\left( \frac{\pi }{6} \right)=7\] \[{{e}^{\int_{{}}^{{}}{pdx}}}={{e}^{-\int_{{}}^{{}}{\tan \,xdx}=}}{{e}^{\ell n\,\cos x}}=\cos x\] \[y.\cos x=\int_{{}}^{{}}{6x\sec x\cos xdx}\] \[y.\cos x=\frac{6{{x}^{2}}}{2}+C\] \[y=3{{x}^{2}}\sec x+C\sec x\] 0\[=3.\frac{{{\pi }^{2}}}{9}.(2)+C(2)\] \[2C=\frac{-2{{\pi }^{2}}}{3}\Rightarrow \] \[y(\pi /6)=3.\frac{{{\pi }^{2}}}{36}.\left( \frac{2}{\sqrt{3}} \right)+\left( \frac{2}{\sqrt{3}} \right).\left( -\frac{{{\pi }^{2}}}{3} \right)\] \[\Rightarrow y=-\frac{{{\pi }^{2}}}{2\sqrt{3}}\]
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