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Explain solution RD Sharma class 12 chapter Differential Equations exercise 21.7 question 24 maths

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Answer: \sec y=-2 \cos x+c

Hint: Separate the terms of x and y and then integrate them.

Given: \tan y \frac{d y}{d x}=\sin (x+y)+\sin (x-y)

Solution: \tan y \frac{d y}{d x}=\sin (x+y)+\sin (x-y)

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

        =\frac{2 \sin x \cos y}{\tan y}

        \begin{aligned} &\Rightarrow \frac{\tan y}{\cos y} d y=2 \sin x d x \\\\ &\Rightarrow \frac{\sin y}{\cos ^{2} y} d y=2 \sin x d x \end{aligned}

          Integrating both sides

        \Rightarrow \int \frac{\sin y}{\cos ^{2} y} d y=\int 2 \sin x d x

        \begin{aligned} &\Rightarrow \sec y=-2 \cos x+c \\\\ &\Rightarrow \sec y=-2 \cos x+c \end{aligned}

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