If y = y(x) and (2 + sin x)/(y + 1) (dy/dx ) = -cos x, y(0) = 1, then y ((π)/2 ) equals

If y = y(x) and (2 + sin x)/(y + 1) (dy/dx ) = -cos x, y(0)… — JEE Main 2004 Mathematics PYQ

AIEEE 2004

y = y(x)

and

\dfrac{2 + \sin x}{y + 1} \left(\dfrac{dy}{dx}\right) = -\cos x

y(0) = 1

, then

y\left(\dfrac{\pi}{2}\right)

equals

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

Show answer and solution

Correct option: A

Solution

Separate variables: dy/(y+1) = -cos x dx/(2+sin x).

Integrate both sides: ln|y+1| = -ln|2+sin x| + C, so (y+1)(2+sin x) = K.

Apply y(0)=1: (1+1)(2+0) = K → K = 4.

At x=π/2: (y+1)(2+1) = 4 → y+1 = 4/3 → y = 1/3.

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