A) \[{{e}^{x}}(\sin y+\cos y)=c\]
B) \[{{e}^{x}}\sin y=c\]
C) \[{{e}^{x}}=c\cos y\]
D) \[{{e}^{x}}=c\sin y\]
E) \[{{e}^{x}}\cos y=c\]
Correct Answer: E
Solution :
The given differential equation is \[{{e}^{x}}\cos ydx-{{e}^{x}}\sin y\,dy=0\] On integrating both sides \[\Rightarrow \] \[\int{\tan y\,dy=\int{1}\,dx}\] \[\Rightarrow \] \[x=log\text{ }sec\text{ }y+log\text{ }c\] \[\Rightarrow \] \[{{e}^{x}}=\sec y\,c\] \[\Rightarrow \] \[{{e}^{x}}\cos y\,=c\]
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