# 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.

• What have you tried so far? – Klangen May 24 at 12:41
• @Klangen I tried to multiply by $x$ the first equation and by $y$ the second one – Егор Пономарёв May 24 at 12:44
• Perhaps, the coefficient of $y$ in the second Eq. could be +. – Dr Zafar Ahmed DSc May 24 at 13:03
• @DrZafarAhmedDSc, thanks, I'll try this out – Егор Пономарёв May 24 at 13:07

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