A) \[{{e}^{y}}=2x+c\]
B) \[{{e}^{-y}}=2x+c\]
C) \[{{e}^{y}}(x+1)=2x+c\]
D) \[{{e}^{y}}(x+1)+c\]
E) \[{{e}^{y}}(x+1)=x+c\]
Correct Answer: C
Solution :
\[(1+x)\frac{dy}{dx}+1=2{{e}^{-y}}\] \[\Rightarrow \] \[{{e}^{y}}(1+x)\frac{dy}{dx}+{{e}^{y}}=2\] \[\Rightarrow \] \[{{e}^{y}}(1+x)dy=(2-{{e}^{y}})dx\] \[\Rightarrow \] \[{{e}^{y}}(1+x)dy+{{e}^{y}}dx=2\,dx\] \[\Rightarrow \] \[d({{e}^{y}}(1+x))=2\,dx\] On integrating both sides, we get \[{{e}^{y}}(1+x)=2x+c\]
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