The solution of the differential equation ydx + (x-y^2)dy = 0 is

The correct answer is Option (3) → $3xy-y^3 = C$: C is an arbitrary constant

Given differential equation: $y\,dx+(x-y^{2})\,dy=0$

Rearrange as a linear equation for $x$ in variable $y$:

$\frac{dx}{dy}= -\frac{x-y^{2}}{y}$

$\frac{dx}{dy}+\frac{1}{y}x = y$

Integrating factor = $e^{\int \frac{1}{y}\,dy}=e^{\ln y}=y$

Multiply both sides by the integrating factor:

$y\frac{dx}{dy}+x = y^{2}$

$\frac{d}{dy}(xy)=y^{2}$

Integrate w.r.t. $y$:

$xy=\frac{y^{3}}{3}+C$

Hence the general solution is:

$xy-\frac{y^{3}}{3}=C$

The general solution of the differential equation is $xy-\frac{y^{3}}{3}=C$.