I have to solve this system of differential equations :

$$ \begin{align}\dot x &= 2000 - 3xy -2x\\ \dot y &= 3xy - 6y\\ \dot z &= 4y - 2z\end{align} $$

1.Which steps are required to solve it?
The only method I know (so far) is separation of variables, which I (probably) can't use. +
2.How do I solve it with matlab?

  • Is it an independent variable, say time etc., that $x,y,z$ are differentiated with respect to? Also you can use Simulink to solve it numerically. – user13838 Oct 20 '11 at 10:07
  • yes... all variables are with respect to time :) – fragant1996 Oct 20 '11 at 10:08
  • 2
    It's a nonlinear system, so don't expect a nice closed form solution that you can find by hand. A similar example, whose solutions have extremely complex behaviour, is the Lorentz system: en.wikipedia.org/wiki/Lorenz_attractor. – Hans Lundmark Oct 20 '11 at 10:54

The solutions with $x$ and $y$ constant are $x=1000, y=0, z = C e^{-2t}$ and $x=2, y=998/3, z = 1996/3 + C e^{-2t}$. You won't find closed-form solutions other than those, but you can get qualitative information from a phase-plane analysis in the $x,y$ variables.

I got one set of solution as

$$ \begin{align*} x(t) &= \frac{4(k+1)}{3k}\\ y(t) &= \frac{2(749k-1)}{3(k+1)}\\ z(t) &= e^{-2t}+\frac{4(749k-1)}{3(k+1)} \end{align*} $$

where $k$ is a constant

First solve $\dot z = 4y - 2z$

  • I don't think that's right unless $k=2$ or $k=1/749$. – Robert Israel Feb 28 '12 at 21:35

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