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 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 $$\log (x^2+y^2)=2t+C \Rightarrow x^2+y^2=D~ e^{2t}.$$