Answer to Question #245184 in Differential Equations for Niddhi

Question

dx/(x^2-y z) =dy/(y^2-zx)=dz/(z^2-xy)

Answer 1

$\dfrac{dx }{x^2-yz}=\dfrac{dy}{y^2-zx}=\dfrac{dz}{z^2-xy}$

Taking the multipliers $y, z, x,$ then the ratio,

$\dfrac{ydx+zdy+xdz }{x^2y-y^2z+y^2z-z^2x+xz^2-x^2y}$ $=\dfrac{ydx+zdy+xdz }{0}$

Hence

$ydx+zdy+xdz=0$

Integrate

$xy+yz+zx=c_1$

Taking the multipliers $xy, xz, x^2,$ then the ratio,

$\dfrac{xydx+xzdy+x^2dz }{x^3y-xy^2z+xy^2z-z^2x^2+x^2z^2-x^3y}$ $=\dfrac{xydx+xzdy+x^2dz }{0}$

Hence

$xydx+xzdy+x^2dz=0$

Integrate

$\dfrac{x^2y}{2}+xyz+x^2z=\dfrac{c_2}{2}$ $x^2y+2xyz+2x^2z=c_2$

Therefore the solutin is

$\Phi(xy+yz+zx,x^2y+2xyz+2x^2z )=0$

where $\Phi$ is an arbitrary function.