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$The solution of differential equation e^{- x} (y + 1) d y + (\cos^{2} x - \sin 2 x) y d x = 0, if y (0) = 1, is$

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a

y+1+excos2x=2

b

y+logy=excos2x

c

logy+1+excos2x=1

d

y+logy+excos2x=2

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detailed solution

Correct option is D

Given that e−x(y+1)dy+cos2⁡x−sin⁡2xydx=0⇒y+1ydy=−excos2⁡x−sin⁡2xdx Integrating on both sides ∫1+1ydy=−∫excos2⁡x−sin⁡2xdx⇒y+log⁡y=−excos2⁡x+c…….. 1) ∵∫exf(x)+f′(x)dx=exf(x) As y(0)=11+log⁡(1)=−e0cos2⁡0+c⇒y+log⁡y+excos2⁡x=2

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The general solution of the differential equation $(1 + y^{2}) d x + (1 + x^{2}) d y = 0$ is

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