A) \[\frac{x}{y}={{e}^{{{x}^{3}}}}+c\]
B) \[\frac{y}{x}={{e}^{x}}+c\]
C) \[xy={{e}^{{{x}^{2}}}}+c\]
D) \[xy\,{{e}^{{{x}^{3}}}}=c\]
E) \[xy={{e}^{{{x}^{3}}}}+c\]
Correct Answer: A
Solution :
\[y\,dx-x\,dy-3{{x}^{2}}{{y}^{2}}{{e}^{{{x}^{3}}}}dx=0\] \[\Rightarrow \] \[\frac{y\,dx-x\,dy}{{{y}^{2}}}-3{{x}^{2}}{{e}^{{{x}^{3}}}}dx=0\] \[\Rightarrow \] \[d\left( \frac{x}{y} \right)-d({{e}^{{{x}^{3}}}})=0\] On integrating both sides, we get \[\frac{x}{y}-{{e}^{{{x}^{3}}}}=c\] \[\Rightarrow \] \[\frac{x}{y}={{e}^{{{x}^{3}}}}+c\]
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