A) \[x+y\text{ }sin\text{ }x=c\]
B) \[x+y\text{ }cos\text{ }x=c\]
C) \[y=x(sin\text{ }x+cos\text{ }x)+c\]
D) \[y\text{ }sin\text{ }x=x+c\]
E) \[y\text{ }co{{s}^{2}}\text{ }x+\text{ }x=c\]
Correct Answer: D
Solution :
\[\frac{dy}{dx}+y\cot \,x=\,\cos ec\,x\] \[IF={{e}^{\int{\cot x}\,dx}}={{e}^{\log \,\sin x}}=\sin x\] \[\therefore \]Solution is \[y\text{ }sin\text{ }x=\int{sin\text{ }x}\text{ }cosec\text{ }x\text{ }dx\] \[\Rightarrow \] \[y\text{ }sin\text{ }x=x+c\]
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