Differential Equations Question 59
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} ) $
Show Answer
Answer:
Correct Answer: B
Solution:
$ \frac{dy}{dx}+\sin ( \frac{x+y}{2} )=\sin ( \frac{x-y}{2} ) $
Therefore $ \frac{dy}{dx}=\sin ( \frac{x-y}{2} )-\sin ( \frac{x+y}{2} ) $
Therefore $ \frac{dy}{dx}=-2\sin ( \frac{y}{2} ).\cos ( \frac{x}{2} ) $
Therefore $ 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}} $ .
Therefore $ \frac{\log \tan \frac{y}{4}}{1/2}=-\frac{2\sin ( x/2 )}{1/2}+c $
Therefore $ \log (\tan \frac{y}{4})=c-2\sin (x/2) $ .
BETA
