# How to deal with homogenous differential equation system?

How to deal with homogenous differential equation system??

## Homework Statement

s'[t] == (-3 s[t])/580, m'[t] == (-19 m[t])/590,
h'[t] == (3 s[t])/580 + (19 m[t])/590 - (2 h[t])/25,
e'[t] == (2 h[t])/25 - (85 e[t])/116,
o'[t] == (85 e[t])/116 - (33 o[t])/131

## Homework Equations

it is a system to clean the Great lake that five lakes linked together

## The Attempt at a Solution

and I know that the solution for s and m are 2900*E^(-3 t/580) and 1180*E^(-19 t/590)
as s=2900, m=1180. and h=850, e=116, o=393

I really want someone can tell me any math software can work with this or any way to do it

HallsofIvy
$$\begin{pmatrix}s \\ m \\ h \\ e \\ 0\end{pmatrix}'= \begin{pmatrix}-\frac{3}{580} & 0 & 0 & 0 & 0 \\ 0 & -\frac{19}{580} & 0 & 0 & 0 \\ \frac{3}{580} & \frac{19}{580} & -\frac{2}{25} & 0 & 0 \\ 0 & 0 & \frac{2}{25} & -\frac{85}{116} & 0 \\ 0 & 0 & 0 & \frac{85}{116} & -\frac{33}{131}\end{pmatrix}\begin{pmatrix}s \\ m \\ h \\ e \\ 0\end{pmatrix}$$