The general solution of the differential equation $ \frac{d y}{d x}+\sin \left(\frac{x+y}{2}\right)=\sin \left(\frac{x-y}{2}\right) $ is (where $ c $ is an arbitrary constant)

Question

The general solution of the differential equation $ \frac{d y}{d x}+\sin \left(\frac{x+y}{2}\right)=\sin \left(\frac{x-y}{2}\right) $ is (where $ c $ is an arbitrary constant), $ \ln \tan \left(\frac{y}{2}\right)=c-2 \sin x $, $ \ln \tan \left(\frac{y}{4}\right)=c-2 \sin \left(\frac{x}{2}\right) $, $ \ln \tan \left(\frac{y}{2}+\frac{\pi}{4}\right)=c-2 \sin x $, $ \ln \tan \left(\frac{y}{4}+\frac{\pi}{4}\right)=c-2 \sin \left(\frac{x}{2}\right) $

MARKS App · Accepted Answer

Given equation d y d x + sin x + y 2 = sin x - y 2 ⇒ d y d x = sin x - y 2 - sin x + y 2 ⇒ d y d x = - 2 sin y 2 cos x 2 ⇒ c o s e c y 2 d y = - 2 cos x 2 d x On integrating both sides, we get ∫ c o s e c y 2 d y = - ∫ 2 cos x 2 d x ⇒ ln tan y 4 1 2 = − 2 sin x 2 1 2 + c ⇒ ln tan y 4 = c − 2 sin x 2

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