A) \[\frac{{{y}^{2}}}{2}+{{e}^{-x/y}}=k\]
B) \[\frac{{{x}^{2}}}{2}+{{e}^{-x/y}}=k\]
C) \[\frac{{{x}^{2}}}{2}+{{e}^{x/y}}=k\]
D) \[\frac{{{y}^{2}}}{2}+{{e}^{x/y}}=k\]
Correct Answer: A
Solution :
\[y\,{{e}^{-x/y}}dx-(x{{e}^{-x/y}}+{{y}^{3}})dy=0\] \[{{e}^{-x/y}}(ydx-xdy)={{y}^{3}}dy\] Þ \[{{e}^{-x/y}}\frac{(ydx-xdy)}{{{y}^{2}}}=ydy\] \[{{e}^{-x/y}}d\left( \frac{x}{y} \right)=ydy\]. Integrating both sides, we get \[k-{{e}^{-x/y}}=\frac{{{y}^{2}}}{2}\] Þ \[\frac{{{y}^{2}}}{2}+{{e}^{-x/y}}=k\]
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