The solution of the equation dydx=sin⁡(x−y) is

dydx=sin⁡(x−y)Putting u=x−y, we have dudx=1−dydx. The given equation can be written as 1−dudx=sin⁡u⇒∫du1−sin⁡ u=dx⇒tan⁡ u+sec⁡ u=x+C⇒tan ⁡(x−y)+sec⁡ (x−y)=x+C.

The solution of the equation dydx=sin⁡(x−y) is

dydx=sin⁡(x−y)Putting u=x−y, we have dudx=1−dydx. The given equation can be written as 1−dudx=sin⁡u⇒∫du1−sin⁡ u=dx⇒tan⁡ u+sec⁡ u=x+C⇒tan ⁡(x−y)+sec⁡ (x−y)=x+C.

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The solution of the equation $\frac{d y}{d x} = \sin (x - y)$ is

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$y + \cot (\frac{x - y}{2}) = c$

none of these

$x + \cot (\frac{x - y}{2}) = c$

$x + \tan (\frac{x - y}{2}) = c$

answer is D.

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Detailed Solution

$\frac{d y}{d x} = \sin (x - y)$

Putting $u = x - y$ , we have $\frac{d u}{d x} = 1 - \frac{d y}{d x}$ .

The given equation can be written as $1 - \frac{d u}{d x} = \sin u$

$\begin{array}{l} \Rightarrow \int \frac{d u}{1 - \sin u} = d x \\ \Rightarrow \tan u + \sec u = x + C \\ \Rightarrow \tan (x - y) + \sec (x - y) = x + C . \end{array}$