Vector components as functions in DSolve

[Dr. Wolfgang Hintze]

I would like to generalize to an arbitrary number L of dimensions the
following simple procedure in two dimensions:

Defining a 2x2 matrix, e.g.

In[66]:=
A = {{0, -1}, {1, 0}};

and a vector of two functions of time t

In[67]:=
r[t] = {x[t], y[t]};

I put up a system of ordinary differential equations

In[77]:=
equ = D[r[t], t] == -A . r[t]

Out[77]=
{Derivative[1][x][t], Derivative[1][y][t]} ==
{y[t], -x[t]}

and solve it using DSolve

In[78]:=
DSolve[equ, r[t], t]

Out[78]=
{{x[t] -> C[1]*Cos[t] + C[2]*Sin[t],
y[t] -> C[2]*Cos[t] - C[1]*Sin[t]}}

The generalization of the vector r[t] should be something like

In[87]:=
L = 5; s = Array[f[t], {L}]

Out[87]=
{f[t][1], f[t][2], f[t][3], f[t][4], f[t][5]}

or

In[88]:=
L = 5; s = Array[f, {L}][t]

Out[88]=
{f[1], f[2], f[3], f[4], f[5]}[t]

I would then like to use

In[65]:=
DSolve[D[s, t] == A . s, s, t]

but nothing of the kind will work.

My question is simple: how do I treat a system of ordinary differential
equations of the form

d(Vector of functions of t)/dt
= function of the vector of functions of t)

in Mathematica?

Any hints welcome.

Wolfgang

[Bob Hanlon]

L=3; s = f[#][t]& /@ Range[L];

A = Table[Random[Integer, {-1,1} ], {L},{L}]

{{0, 0, 0}, {0, -1, 1}, {-1, 0, 1}}

equ = Thread[D[s,t]==A.s];

DSolve[equ, s,t]

{{f[1][t] -> C[1], f[2][t] -> ((-(1/2))*C[1]*(-1 + E^t)^2)/
E^t + C[2]/E^t + ((1/2)*(-1 + E^(2*t))*C[3])/E^t,
f[3][t] -> (1 - E^t)*C[1] + E^t*C[3]}}

Bob Hanlon

In article <beb6a7$66o$1...@smc.vnet.net>, "Dr. Wolfgang Hintze" <w...@snafu.de>
wrote:

[Selwyn Hollis]

Wolfgang,

My DETools package extends DSolve and NDSolve so that they understand
*linear* systems in vector form. See

http://www.math.armstrong.edu/faculty/hollis/mmade.

Actually, I need to take that out for version 5.0, since it's
apparently not needed. See

http://www.wolfram.com/products/mathematica/newin5/numeric/ndsolve.html

-----
Selwyn Hollis
http://www.math.armstrong.edu/faculty/hollis

> Any hints welcome.
>
> Wolfgang
>
>