A) \[c{{e}^{-x}}+\frac{2}{3}{{e}^{2x}}\]
B) \[(1+x){{e}^{-x}}+\frac{2}{3}{{e}^{2x}}+c\]
C) \[c{{e}^{-x}}+\frac{2}{3}{{e}^{2x}}+c\]
D) \[{{e}^{-x}}+\frac{2}{3}{{e}^{2x}}+c\]
Correct Answer: A
Solution :
Given,\[\frac{dy}{dx}+y=2{{e}^{2x}}\] \[IF={{e}^{\int{1\,\,dx}}}={{e}^{x}}\] \[\therefore \]Required solution is \[y(IF)=\int{2{{e}^{2x}}\cdot {{e}^{x}}\,\,dx}\] \[\Rightarrow \] \[y\,\,{{e}^{x}}=2\int{{{e}^{2x}}}{{e}^{x}}dx\] \[=2\int{{{e}^{3x}}dx}\] \[\Rightarrow \] \[y=\frac{2}{3}{{e}^{3x}}{{e}^{-x}}+c{{e}^{-x}}\] \[\Rightarrow \] \[y=\frac{2}{3}{{e}^{2x}}+c{{e}^{-x}}\]
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