A) \[y+{{e}^{x}}=c\]
B) \[y-{{e}^{-x}}=c\]
C) \[y+{{e}^{-x}}=c\]
D) \[y-{{e}^{x}}=c\]
Correct Answer: C
Solution :
[c] Consider the given differential equation \[\ln \,\left( \frac{dy}{dx} \right)+x=0\Rightarrow ln\left( \frac{dy}{dx} \right)=-x\] \[\Rightarrow \frac{dy}{dx}={{e}^{-x}}\] On separating the variables, we get \[dy={{e}^{-x}}dx,\] On integrating both sides, we get \[\int{dy=\int{{{e}^{-x}}dx}}\] \[\Rightarrow y=\frac{{{e}^{-x}}}{-1}+C=-{{e}^{-x}}+C\Rightarrow y+{{e}^{-x}}=C\]
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