If `y=y(x)` and `((2+sinx)/(y+1))dy/dx=-cosx, y(0)=1`, then `y(pi/2)` equals (A) `1/3` (B) `2/3` (C) `-1/3` (D) `1`

Options

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

Answer

A

Explanation

`1/(y+1)dy=-(cosx)/(2+sinx)dx`
Integrating, we get
`log(y+1)+logk+log(2+sinx)=0`
`therefore k(y+1)(2+sinx)=1` when `x=0, y=1` where k is constant.
`therefore 4k=1` or `k=1//4`
`therefore (y+1)(2+sinx)=4`
Now, put `x=pi/2`
`therefore (y+1)3=4 therefore y=1//3`