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 :
It can be also solved by comparing with the linear equation \[\frac{dy}{dx}+Py=Q\] The integrating factor, \[I.F={{e}^{\int_{{}}^{{}}{1.dx}}}={{e}^{x}}\] Therefore, \[I.F=\int_{{}}^{{}}{2{{e}^{2x}}.I.F+C}\] \[y.{{e}^{x}}=\int_{{}}^{{}}{2{{e}^{2x}}.{{e}^{x}}+C}\] \[y.{{e}^{x}}=2\int_{{}}^{{}}{{{3}^{3x}}+C}=\frac{2}{3}{{e}^{3x}}+C\Rightarrow y=\frac{2{{e}^{x}}}{3}+c{{e}^{-x}}\]
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