What is a general solution to the differential equation dy/dx=(2x)/(y+x^2y)^2?

Aug 9, 2016

$y = \sqrt[3]{C - \frac{3}{1 + {x}^{2}}}$

Explanation:

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

this is separable

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{y} ^ 2 \cdot \frac{2 x}{1 + {x}^{2}} ^ 2$

${y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{1 + {x}^{2}} ^ 2$

$\implies {y}^{3} / 3 = - \frac{1}{1 + {x}^{2}} + C$

${y}^{3} = C - \frac{3}{1 + {x}^{2}}$

$y = \sqrt[3]{C - \frac{3}{1 + {x}^{2}}}$