# The solution of differential equation $x^{2}ydx-\left ( x^{3}+y^{2} \right )dy=0$ is Option 1) $\frac{1}{3}\frac{x^{3}}{y^{3}}+log \:y=C$ Option 2) $\frac{1}{3}\frac{x^{3}}{y^{3}}-log \:y=C$ Option 3) $\frac{x^{3}}{y^{3}}+log \:y=-C$ Option 4) None of these

The solution of differential equation $x^{2}ydx-\left ( x^{3}+y^{2} \right )dy=0$ is

$x^{2}ydx-(x^{3}+y^{3})dy=0\\ \therefore \frac{dx}{dy}=\frac{x^{3}+y^{3}}{x^{2y}}\\$

Divide by y3

$\frac{dx}{dy}=\frac{1+(\frac{x}{y})^{3}}{(\frac{x}{y})^{2}}$

Now let $\frac{x}{y}=v=x=vy\\ \frac{dx}{dy}=\frac{1+V^{3}}{V^{2}}\\ y\frac{dv}{dy}=\frac{1+V^{3}}{V^{2}}-V=\frac{1+V^{3}-V^{3}}{V^{2}}=\frac{1}{V^{2}}\\$

$\int v^{2}dv=\int \frac{dy}{y}\\ \therefore \frac{V^{3}}{3}=logy+c\\ \frac{1}{3}.\frac{x^{3}}{y^{3}}-logy=c$

Option 1)

$\frac{1}{3}\frac{x^{3}}{y^{3}}+log \:y=C$

This solution is incorrect

Option 2)

$\frac{1}{3}\frac{x^{3}}{y^{3}}-log \:y=C$

This solution is correct

Option 3)

$\frac{x^{3}}{y^{3}}+log \:y=-C$

This solutioin is incorrect

Option 4)

None of these

This solution is incorrect

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