A) \[2y={{e}^{2x}}+c\]
B) s\[2y{{e}^{x}}={{e}^{2}}+c\]
C) \[2y{{e}^{x}}={{e}^{2x}}+c\]
D) \[2y{{e}^{2x}}=2{{e}^{x}}+c\]
Correct Answer: C
Solution :
Given, differential equation is \[\frac{dy}{x}+y={{e}^{x}},\] which is of the form \[\frac{dy}{dx}+Py=Q\] Here, \[P=1,\,\,Q={{e}^{x}}\] \[\therefore \] \[IF={{e}^{\int{Pdx}}}={{e}^{\int{1dx}}}={{e}^{x}}\] Now, solution is \[y.IF\,=\,\int{Q.\,IF\,\,dx}\] \[\Rightarrow \] \[y{{e}^{x}}=\int{{{e}^{2x}}\,\,dx}\] \[\Rightarrow \] \[y{{e}^{x}}=\frac{{{e}^{2x}}}{2}+c\] \[\Rightarrow \] \[2y{{e}^{x}}={{e}^{2x}}+c\]
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