A) \[{{x}^{2/3}}+{{y}^{2/3}}=c\]
B) \[{{x}^{1/3}}+{{y}^{1/3}}=c\]
C) \[{{y}^{2/3}}-{{x}^{2/3}}=c\]
D) \[{{y}^{1/3}}-{{x}^{1/3}}=c\]
Correct Answer: C
Solution :
\[\frac{dy}{dx}={{\left( \frac{y}{x} \right)}^{1/3}}\] Þ \[\frac{dy}{dx}=\frac{{{y}^{1/3}}}{{{x}^{1/3}}}\] Þ \[\frac{dy}{{{y}^{1/3}}}=\frac{dx}{{{x}^{1/3}}}\] Integrating both sides, \[\frac{{{y}^{2/3}}}{2/3}=\frac{{{x}^{2/3}}}{2/3}+c\] \[\frac{3}{2}\cdot {{y}^{2/3}}=\frac{3}{2}{{x}^{2/3}}+c\] Þ \[{{y}^{2/3}}-{{x}^{2/3}}=c\].
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