This is an solution of an partial differential equation by Lagrange method of multipliers.
My Question is as indicated by arrows we get zero at the denominator and "it's get transfered to the other side of the equation which equates the whole equation to zero."
How is this possible? (X/0) is not defined or infinity. It's doing like " x/0 = y x = 0*y x = 0 " which is wrong.
It's just a notation. Note that you can find a constant of integration this way without the zero at the denominator:
$$\dfrac {dx}{y+z}=\dfrac {dy}{-(x+z)}=\dfrac {dz}{x-y}$$ $$\dfrac {dx}{y+z}=\dfrac {d(y+z)}{-(y+z)}$$ $$\dfrac {dx}{1}=\dfrac {d(y+z)}{-1}$$ $$dx=-d(y+z)$$ $$x+y+z=c_1$$ it gives the same result.
For th second constant of integration $$\dfrac {dx}{y+z}=\dfrac {dy}{-(x+z)}=\dfrac {dz}{x-y}$$ $$\dfrac {xdx}{xy+xz}=\dfrac {ydy}{-(yx+zy)}=\dfrac {dz}{x-y}$$ $$\dfrac {xdx+ydy}{xz-zy}=\dfrac {dz}{x-y}$$ $$\dfrac {xdx+ydy}{z}=\dfrac {dz}{1}$$ $$xdx+ydy=zdz$$ Integration gives: $$x^2+y^2=z^2+c_2$$