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Given the continuous time dynamical system with the rule depending on time: $$x'=3x+(2-t)y\qquad y'=xy-t$$ create a new system which is equivalent to the above system for which the rule does not depend on $t$.

My solution so far: I recognize that this can be written in matrix for as in $Y'=AY$

|Y'| = |x'| = |3   2t| * |x|

       |y'|   |y -t/y|   |y|

I'm stuck here because I get the feeling my A shouldn't include any variables, so I tried to solve the DEs. I start with $y'=xy-t$:

$y'=xy-t$

$y'-xy=-t$

multiply by $u(t)=e^{-xt}$

d(ye^-(xt))/dt=-te^-(xt)

ye^-(xt)=[e^(xt)(1-xt)]/x^2

y=[e^(2xt)(1-xt)]/x^2

Then I'm stuck here but I'm pretty sure this isn't the way to go.

Did
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Mattszo
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  • How do you define "equivalent" here? – Did Sep 23 '15 at 08:27
  • I believe when you take the integral of $-te^{-xt}$ you consider $x$ independent of $t$. Is this correct? Also can you explain what you mean by a rule that does not depent on $t$? As far as I understand a system of differential equations is supposed to give a solution of the form $\phi(t):=(x(t),y(t))$, which is also called a flow or a continuous dynamical system. So finding an equivalent dynamical system means to find a function $y=f(x)$ such that $\phi$ and the graph of $f$ coincide, yes? – Alp Uzman Sep 23 '15 at 09:10
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    just create another variable $z=t$, with $z'=1$. Replace all occurences of $t$ in your equations with $z$. – nonlinearism Sep 23 '15 at 13:40
  • okay I think that I was headed in the wrong direction. I now realize that I can't separate it into y'=Ay because it is not linear, and they just want me to produce an equivalent system, so I don't think I need to do any DE solving here. I replaced all 't's with 'z's and I get the same thing ... but with z. Is that really all they're looking for ? @nonlinearism – Mattszo Sep 23 '15 at 22:32

1 Answers1

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Although the question is old, I would like to propose a solution. I believe what the question is asking is finding an equation which does not depend on t. Thus you can replace t with t=-y'+xy (obtained from second equation) in the first equation and you obtain a time independent equation.

tempx
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