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.

  • $\begingroup$ What have you tried so far? $\endgroup$ – Klangen May 24 at 12:41
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    $\begingroup$ @Klangen I tried to multiply by $x$ the first equation and by $y$ the second one $\endgroup$ – Егор Пономарёв May 24 at 12:44
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    $\begingroup$ Perhaps, the coefficient of $y$ in the second Eq. could be +. $\endgroup$ – Dr Zafar Ahmed DSc May 24 at 13:03
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    $\begingroup$ @DrZafarAhmedDSc, thanks, I'll try this out $\endgroup$ – 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 $$\log (x^2+y^2)=2t+C \Rightarrow x^2+y^2=D~ e^{2t}.$$


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