A) \[\log x+\frac{x}{y}\]
B) \[\log x+\frac{{{y}^{3}}}{x}=k\]
C) \[\log x-\frac{x}{{{y}^{3}}}=k\]
D) \[\log xy-{{y}^{3}}=k\]
Correct Answer: B
Solution :
\[xdx-{{y}^{3}}dx+3x{{y}^{2}}dy=0\] Put \[{{y}^{3}}=t\] \[\Rightarrow \] \[dt=3{{y}^{2}}dy\] \[x\,dx-tdx+xdt=0\] Þ \[xdx+xdt-tdx=0\] Þ \[\frac{dx}{x}+d\left( \frac{t}{x} \right)=0\] On integration, we get \[\log x+\frac{t}{x}=k\] Þ \[\log x+\frac{{{y}^{3}}}{x}=k\].
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