# Solve the following system of differential equations <mtable columnalign="right left right left

Solve the following system of differential equations
$\begin{array}{rl}\stackrel{˙}{x}& =2000-3xy-2x\\ \stackrel{˙}{y}& =3xy-6y\\ \stackrel{˙}{z}& =4y-2z\end{array}$
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Zichetti4b
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.

Jovany Clayton
$\begin{array}{rl}x\left(t\right)& =\frac{4\left(k+1\right)}{3k}\\ y\left(t\right)& =\frac{2\left(749k-1\right)}{3\left(k+1\right)}\\ z\left(t\right)& ={e}^{-2t}+\frac{4\left(749k-1\right)}{3\left(k+1\right)}\end{array}$
where $k$ is a constant
First solve $\stackrel{˙}{z}=4y-2z$