If `y=y(x)` and `(2+sinx)/(y+1) (dy)/(dx)=-cosx, y(0)=1`, then `y(pi//2)` equals

Options

A. `(1)/(3)`
B. `(2)/(3)`
C. `-(1)/(3)`
D. 1

Answer

A

Explanation

We have,
`(2sin x)/(y+1)(dy)/(dx)=-cosx`
`rArr" "(1)/(y+1)dy=-(cosx)/(2+sinx)dx`
`rArr" "(1)/(y+1)dy=-(cosx)/(2+sinx)dx`
`rArr" "int" "(1)/(y+1)dy=-int" "(cosx)/(2+sinx)dx`
`rArr" "log(y+1)=-log(2+sinx)+logC`
`rArr" "y+1=(C)/(2+sinx)`
Putting x = 0 and y = 1 in (i), we get
`2=(C)/(2) rArr C=4`
Putting C = 4 (i)., we get `y+1=(4)/(2+sinx)`
Putting `x=(pi)/(2)`, we get
`y+1=(4)/(3)rArr y=(1)/(2)rArr ((pi)/(2))=(1)/(3)`