A) \[y={{e}^{-x}}(x-1)\]
B) \[y=x{{e}^{-x}}\]
C) \[y=x{{e}^{-x}}+1\]
D) \[y=(x+1){{e}^{-x}}\]
E) \[y=x{{e}^{x}}\]
Correct Answer: B
Solution :
The given differential equation is \[\frac{dy}{dx}+y={{e}^{-x}}\] On comparing with\[\frac{dy}{dx}+Py=Q,\]we get \[P=1\]and \[Q={{e}^{-x}}\] \[\therefore \] \[IF={{e}^{\int{P}\,dx}}={{e}^{x}}\] \[\therefore \]Required solution is \[y{{e}^{x}}=\int{{{e}^{x}}.{{e}^{-x}}}dx+c\] \[\Rightarrow \] \[y{{e}^{x}}=x+c\] At\[y=0\]and\[x=0\] \[\Rightarrow \] \[c=0,\] \[\therefore \] \[y=x{{e}^{-x}}\]
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