the total differential equation $(y^2z - y^3 +x^2y)dx - (x^2z +x^3 -xy^2)dy +(x^2y - xy^2)dz=0$ In solving the total differential equation $$(y^2z - y^3 +x^2y)dx - (x^2z +x^3 -xy^2)dy +(x^2y - xy^2)dz=0$$
Substituting $x =uz$ and $y=vz$ and reached at a position where, i face the following
$$(v^2 - v^3 +u^2v)du - (u^2+u^3-uv^2)dv=0$$
I am stuck here please help!!

Your equation $(v^2 - v^3 +u^2v)du - (u^2+u^3-uv^2)dv=0$ is OK. To continue, see below :

Instead of such a complicated calculus, a little bit of observation would have given the integrating factor $\frac{1}{u^2v^2}$ for $(v^2 - v^3 +u^2v)du - (u^2+u^3-uv^2)dv=0$
$$\left(\frac{1}{u^2}-\frac{v}{u^2}+\frac{1}{v}\right)du+\left(-\frac{1}{v^2}-\frac{u}{v^2}+\frac{1}{u}\right)dv=0$$
$$d\left(-\frac{1}{u}+\frac{v}{u}+\frac{u}{v}+\frac{1}{v}\right)=0$$
$$-\frac{1}{u}+\frac{v}{u}+\frac{u}{v}+\frac{1}{v}=constant$$
$$-\frac{z}{x}+\frac{y}{x}+\frac{x}{y}+\frac{z}{y}=constant$$ 
