0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What have you tried so far? $\endgroup$ – Klangen May 24 at 12:41
  • 1
    $\begingroup$ @Klangen I tried to multiply by $x$ the first equation and by $y$ the second one $\endgroup$ – Егор Пономарёв May 24 at 12:44
  • 2
    $\begingroup$ Perhaps, the coefficient of $y$ in the second Eq. could be +. $\endgroup$ – Dr Zafar Ahmed DSc May 24 at 13:03
  • 1
    $\begingroup$ @DrZafarAhmedDSc, thanks, I'll try this out $\endgroup$ – Егор Пономарёв May 24 at 13:07
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.