How do I solve this system of differential equations?

Question

I have to solve the following 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?

Answer 1

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 $\dot z = 4y - 2z$

Answer 2

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.