Integrating factor for $(x^2-y^2-y)dx-(x^2-y^2-x)dy=0$ I am having trouble finding the integrating factor for turning the below differential equation into an exact one (Tenenbaum and Pollard, exercise 10, problem 6). Any hints and suggestions would extremely helpful and lead me to the solution.

Solve the differential equation :
$$ (x^2-y^2-y)dx - (x^2-y^2-x)dy=0$$

My attempt:
The coefficients of $dx$ and $dy$ are not homogenous functions. Further,
$
\begin{align}
P(x,y) &= x^2 - y^2 - y \\
\frac{\partial P(x,y)}{\partial y}&=-2y-1 \\
Q(x,y) &= -(x^2-y^2-x)\\
\frac{\partial Q(x,y)}{\partial x}&=-(2x-1) \\
\therefore \frac{\partial P}{\partial y} \neq \frac{\partial Q}{\partial x}
\end{align}
$
The given differential equation is not exact. We have :
$\begin{align}
& \frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x} \\
=& -2y-1+(2x-1)\\
=&(2x-2y-2)
\end{align}$.
Moreover,
$\begin{align}
& yQ-xP\\
= & -y(x^2-y^2-x)-x(x^2-y^2-y)\\
= & -x^2 y + y^3 + xy - x^3 + xy^2 + xy\\
= & y^3 - x^3 + xy - xy(x - y - 1)
\end{align}$
It doesn't look like $(\partial P / \partial y - \partial Q / \partial x)/(yQ-xP)$ will be a function of $u=xy$ alone.
Also,
$\begin{align}
& yQ+xP\\
= & -y(x^2-y^2-x)+x(x^2-y^2-y)\\
= & -x^2 y + y^3 + xy + x^3 - xy^2 - xy\\
= & y^3 + x^3 - x^2 y - x y^2 \\
= & (y + x)(y^2 + x^2 - xy) - xy(y + x) \\
= & (y + x)(y^2 + x^2 - 2xy)\\
= & (y + x)(y - x)^2
\end{align}$
It doesn't look like $y^2(\partial P / \partial y - \partial Q / \partial x)/(yQ+xP)$ will be a function of $u=x/y$ or  $x^2(\partial P / \partial y - \partial Q / \partial x)/(yQ+xP)$ will be a function of $u=y/x$  alone.

The part of $\dfrac{\partial P}{\partial y} - \dfrac{\partial Q}{\partial x} = 2(x-y-1)=2(z-1)$ shows that you may assume a new variable $z=x-y$ which which shows the denominator of integrating factor might be of the form 
$$\dfrac{\partial I}{\partial x}Q - \dfrac{\partial I}{\partial y}P = P+Q=z$$
so
$$p(z)=\dfrac{2(z-1)}{z}$$
