A) \[\frac{{{y}^{2}}}{x}-{{x}^{3}}{{y}^{2}}=c\]
B) \[\frac{{{x}^{2}}}{{{y}^{2}}}+{{x}^{3}}{{y}^{3}}=c\]
C) \[\frac{{{x}^{2}}}{y}+{{x}^{3}}{{y}^{2}}=c\]
D) \[\frac{{{x}^{2}}}{3y}-{{x}^{3}}{{y}^{2}}=c\]
Correct Answer: C
Solution :
[c] Rewrite the differential equation as \[(2xydx-{{x}^{2}}dy)+{{y}^{2}}(3{{x}^{2}}{{y}^{2}}dx+2{{x}^{3}}ydy)=0\]Dividing by\[{{y}^{2}}\], we get \[\frac{y2xdx-{{x}^{2}}dy}{{{y}^{2}}}+{{y}^{2}}3{{x}^{2}}dx+{{x}^{3}}2ydy=0\] Or \[d\left( \frac{{{x}^{2}}}{y} \right)+d\left( {{x}^{2}}{{y}^{2}} \right)=0\] Integrating, we get the solution \[\frac{{{x}^{2}}}{y}+{{x}^{3}}{{y}^{2}}=c\]
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