# How to deal with homogenous differential equation system?

1. Nov 17, 2011

### yeland404

How to deal with homogenous differential equation system??

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

3. 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[0]=2900, m[0]=1180. and h[0]=850, e[0]=116, o[0]=393

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

2. Nov 18, 2011

### HallsofIvy

Staff Emeritus
Re: How to deal with homogenous differential equation system??

Okay, since you were able to solve for s and m, put those into the equation for h and it becomes pretty straight forward. Once you know h, you can put that into the equation for e and then put the solution for e into the equation for o.

Or you can write it as the matrix equation
$$\begin{pmatrix}s \\ m \\ h \\ e \\ 0\end{pmatrix}'= \begin{pmatrix}-\frac{3}{580} & 0 & 0 & 0 & 0 \\ 0 & -\frac{19}{580} & 0 & 0 & 0 \\ \frac{3}{580} & \frac{19}{580} & -\frac{2}{25} & 0 & 0 \\ 0 & 0 & \frac{2}{25} & -\frac{85}{116} & 0 \\ 0 & 0 & 0 & \frac{85}{116} & -\frac{33}{131}\end{pmatrix}\begin{pmatrix}s \\ m \\ h \\ e \\ 0\end{pmatrix}$$
Since that is a triangular matrix, its eigenvalues are just the numbers on the main diagonal.

3. Nov 18, 2011

### yeland404

Re: How to deal with homogenous differential equation system??

I know how to do with the X'=AX that x=c1e^λt[u1]....however, what I get is a single equation. I have no idea how to deal with five variables....