Differential-Equations Question 377
Question: The general solution of the differential equation $ \frac{dy}{dx}+\sin ( \frac{x+y}{2} )=\sin ( \frac{x-y}{2} ) $ is
[MP PET 2001]
Options:
A) $ \log \tan ( \frac{y}{2} )=c-2\sin x $
B) $ \log \tan ,( \frac{y}{4} )=c-2\sin ( \frac{x}{2} ) $
C) $ \log \tan ,( \frac{y}{2}+\frac{\pi }{4} )=c-2\sin x $
D) $ \log \tan ( \frac{y}{4}+\frac{\pi }{4} )=c-2\sin ( \frac{x}{2} ) $
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Answer:
Correct Answer: B
Solution:
$ \frac{dy}{dx}+\sin ( \frac{x+y}{2} )=\sin ( \frac{x-y}{2} ) $
Þ $ \frac{dy}{dx}=\sin ( \frac{x-y}{2} )-\sin ( \frac{x+y}{2} ) $
Þ $ \frac{dy}{dx}=-2\sin ,( \frac{y}{2} ),.\cos ,( \frac{x}{2} ) $
Þ $ cosec( \frac{y}{2} ).dy=-2\cos ( \frac{x}{2} ),dx $ Integrating both sides, $ \int{cosec( \frac{y}{2} )dy=-\int{2\cos ( \frac{x}{2} )dx+c}} $ .
Þ $ \frac{\log ,\tan \frac{y}{4}}{1/2}=-\frac{2\sin ( x/2 )}{1/2}+c $
Þ $ \log (\tan \frac{y}{4})=c-2\sin (x/2) $ .
BETA
