Question

Homogeneous Differential equation:
$x{y}^{2}dy-(y^3-2x^3)dx=0$

Answer 1

$x{y}^{2}dy-(y^3-2x^2)dx=0$

$\frac{dy}{dx}=\frac{y^3-2x^3}{x{y}^2}$

Put $y=vx$

$\frac{dy}{dx}=v+x\frac{dv}{dx}$

$v+x\frac{dv}{dx}=\frac{y}{x}-2\frac{x^2}{y^2}=u-\frac{2}{v^2}$

$x\frac{dv}{dx}=\frac{-2}{v^2}$

$v^{2}dv=-2\frac{dx}{x}$

$\frac{v^3}{3}=-2\log x+\log C$

$\frac{v^3}{3}=\frac{c}{x^2}$

$y^3=3cx$