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$The solution of differential equation 2 y \sin x (\frac{d y}{d x}) = 2 \sin x \cos x - y^{2} \cos x at x = \frac{π}{2}, y = 1 is$

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a

y2=sinx

b

y=sin2x

c

y2=cosx+1

d

None

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detailed solution

Correct option is A

Given equation can be rewrite as 2ydydx+y2cot⁡x=dvdx Put y2=v=2ydydx=dvdxdvdx+vcot⁡x=2cos⁡x If e∫cos⁡xsin⁡x=dx=elog⁡sin⁡x=sin⁡x Solution is vsin⁡x+∫2cos⁡xsin⁡xdx+cy2sin⁡x=sin2⁡x+c ; when x=π2,y=11=1+c⇒c=0; y2sin⁡x=sin2x⇒y2=sinx

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