A) \[{{e}^{y}}={{e}^{x}}+\frac{{{x}^{3}}}{3}+c\]
B) \[{{e}^{y}}={{e}^{x}}+2x+c\]
C) \[{{e}^{y}}={{e}^{x}}+{{x}^{3}}+c\]
D) \[y={{e}^{x}}+c\]
Correct Answer: A
Solution :
\[\frac{dy}{dx}={{e}^{x-y}}+{{x}^{2}}{{e}^{-y}}={{e}^{-y}}({{e}^{x}}+{{x}^{2}})\] Þ \[{{e}^{y}}dy=({{x}^{2}}+{{e}^{x}})dx\] Now integrating both sides, we get \[{{e}^{y}}=\frac{{{x}^{3}}}{3}+{{e}^{x}}+c\].
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