Differential Equations Question 72
Question: If $ ( \frac{2+\sin x}{1+y} )\frac{dy}{dx}=-\cos x,\ y(0)=1, $ then $ y( \frac{\pi }{2} ) $ =
[IIT Screening 2004]
Options:
A) 1
B) $ \frac{1}{2} $
C) $ \frac{1}{3} $
D) $ \frac{1}{4} $
Show Answer
Answer:
Correct Answer: C
Solution:
The given differential equation is $ \frac{\cos x}{2+\sin x}dx+\frac{dy}{y+1}=0 $
Therefore $ \log (2+\sin x)+\log (y+1)=\log c $
Therefore $ (y+1)(2+\sin x)=c $
Therefore $ 2\times 2=c $
Therefore $ c=4 $
Thus, $ y+1=\frac{4}{2+\sin x} $
Therefore $ y=\frac{2-\sin x}{2+\sin x} $
Therefore $ y( \frac{\pi }{2} )=\frac{1}{3} $ .
BETA
