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$The solution of the equation \cos^{2} x \frac{d y}{d x} - (\tan 2 x) y = \cos^{4} x, | x | < \frac{π}{4} when y (π / 6) = \frac{3 \sqrt{3}}{8} is$

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a

y=tan2xcos2x

b

y=cot2x.cos2x

c

y=12tan2xcos2x

d

y=12cot2xcos2x

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detailed solution

Correct option is C

dydx−tan⁡2xcos2⁡xy=cos2⁡x I.E. =e-∫tan2xcos2xdxdx=e-∫2tanx1-tan2xsec2x dxput tanx=t, sec2x dx=dt=e∫-2t1-t2dt=elog(1-t2)=1-tan2x Solution of D.E is y(1-tan2x)=∫(1-tan2x)cos2xdxy1-sin2xcos2x=∫1-sin2xcos2xcos2x dxycos2xcos2x=∫cos2x dxycos2xcos2x=sin2x2+C When x=π/6,y=338338cosπ3cos2π6=sinπ32+C3381234=322+C⇒C=0ycos2xcos2x=sin2x2y=12tan⁡2xcos2⁡x

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