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If (2+sinx)(dy)/(dx)+(y+1)cosx=0 and y(0...

If `(2+sinx)(dy)/(dx)+(y+1)cosx=0` and `y(0)=1`, then `y(pi/2)` is equal to

A

`1/3`

B

`-2/3`

C

`-1/3`

D

`4/3`

Text Solution

Verified by Experts

The correct Answer is:
A
We have, `(2+sinx)dy/dx+(y+1)cos x =0`
`rArr dy/dx+(cosx)/(2+sinx)y=(-cosx)/(2+sinx)`
which is a linear differential equation.
`therefore IF=e^(int(cosx)/(2+sinx)dx)=e^(log(2+sinx))=2 +sin x`
`therefore` Required solution is given by
`y cdot (2+sinx)=int(-cosx)/(2+sin x)cdot (2+sinx)dx+C`
`rArr y(2+sinx) =-sinx+C`
Also, `y(0)=1`
`therefore 1(2+sin0)=-sin0+C`
`rArr C =2`
`therefore y=(2-sinx)/(2+sinx) rArr y(pi/2)=(2-sin frac{pi}{2})/(2+sin frac {pi}{2})=1/3`
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