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If y = y ( x ) is the solutio...

If ` y = y ( x ) ` is the solution of differential equation ` sin y (dy ) /(dx ) - cos y = e ^ ( - x ) ` such that ` y ( 0 ) = ( pi ) /(2) ` then ` y (A) ` is equal to

A

`"sin"^(-1)1/e`

B

`"cos"^(-1)1/e`

C

`-"cos"^(-1)1/e`

D

`cos^(-1)(-1/e)`

Text Solution

Verified by Experts

The correct Answer is:
D
sin y `(dy)/(dx)-cos y=e^(-x)`
Put -cos y=t `rArr sin y (dy)/(dx) = (dt)/(dx) rArr (dy)/(dx)+t=e^(-x)`
If =`e^(int dx) =e^x rArr t.e^x = inte^(-x) e^x dx+c rArr -cos y .e^x =x+c`
`therefore y(0)=pi/2 rArr 0=0+c rArr c=0 , cos y = -xe^(-x) rArr y(1)=cos^(-1) (-1/e)`
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