Question

If $(2 + \sin \, x ) \frac{dy}{dx} + (y + 1) \cos \, x = 0$ and $y(0) = 1,$ then $y \left( \frac{\pi}{2} \right)$ is equal to :

• A. $- \frac{2}{3}$
• B. $- \frac{1}{3}$
• C. $ \frac{4}{3}$
• D. $ \frac{1}{3}$

Answer 1

The Correct Option is D

$\left(2+\sin x\right) \frac{dy}{dx}+\left(y+1\right) \cos x = 0$
$y (0) = 1, y \left(\frac{\pi}{2}\right)$ = ?
$\frac{1}{y+1}dy+\frac{\cos x}{2+\sin x} dx = 0$
$In\left|y+1\right|+In\left(2+\sin x\right) = InC$
$(y + 1) (2 + \sin x) = C$
Put $x = 0,\, y = 1$
$(1 + 1) \cdot 2 = C \Rightarrow C = 4$
Now, $(y +1)(2 + \sin x) = 4$
For, $x = \frac{\pi}{2}$
$(y +1)(2 +1) = 4$
$y + 1 = \frac{4}{3}$
$y = \frac{4}{3}-1 =\frac{1}{3}$