Q:

Solve system of differential equations using integrating combination

I am trying to solve this system of differential equations:
$$
\left\{ 
\begin{array}{c}
\dot{x} = -xy^2+x+y \\ 
\dot{y} = -x-y+x^2y 
\end{array}
\right.
$$
I am trying to find integrating combination to solve this equation. Could you please provide it?
I will provide an example:
System:
$$
\left\{ 
\begin{array}{c}
x' = \frac{x}{z} \\ 
y' = -\frac{x}{y} 
\end{array}
\right.
$$
an integrating combination for it:
$$
\frac{dz}{dy} = -\frac{z}{y}
$$
it was get from the system by combinating its participants.

A:

A possibility: when we change $-y$ to $+y$ in Equation (2) such that we have to solve  $$\dot x=-xy^2+x+y~~~(1)~~~~~ \mbox{and}~~~~~~ \dot y=-x+y+x^2y~~~(2).$$
Multiply (1) by $x$ and (2) by $y$ and add both the equations as:
$$x\dot x+y\dot y=x^2+y^2 \Rightarrow \frac{1}{2} \frac{d}{dt} (x^2+y^2)=(x^2+y^2) \Rightarrow \int \frac{d(x^2+y^2)}{x^2+y^2}=2 \int  dt. $$
We get the solution as  $$\log (x^2+y^2)=2t+C \Rightarrow x^2+y^2=D~ e^{2t}.$$

