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MathGroup Archive 2005

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Vector of functions in ordinary differential equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg57109] Vector of functions in ordinary differential equations
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Mon, 16 May 2005 01:29:50 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Considering ordinary differential equations for more than one function 
I'd like to have one differential equation for n functions in terms a 
vector rnv[t] = {x[1,t], x[2,t], ... , x[n,t]} but I did not succeed in 
expressing it in Mathematica.

Any hints are greatly appreciated.

Regards,
Wolfgang

Here are the details:

1) There are no problems if I define the independent functions 
explicitly like x[t], y[t], ... and write down a differential eqaution 
for each component.

2) There are also no problems if the explicit components are collected 
into a vector, e.g. rv[t] := {x[t], y[t]}. Then one differential 
equation for the vector works fine. This is a example for a linear diffeq:

In[103]:=
rv[t_] := {x[t], y[t]}

In[104]:= rv[t]

Out[104]= {x[t], y[t]}

In[105]:= M = {{a, b}, {c, d}}

Out[105]= {{a, b}, {c, d}}

In[108]:= eq = D[rv[t], t] == M . rv[t]

Out[108]= {Derivative[1][x][t], Derivative[1][y][t]} == {a*x[t] + 
b*y[t], c*x[t] + d*y[t]}

In[110]:= sm = DSolve[eq, rv[t], t];

In[74]:= x[t] /. sm

Out[74]= {1/(2*Sqrt[a^2 + 4*b*c ... [snip]

3) Now, in the general case, I'd like to have n functions x[1,t], 
x[2,t], ... , x[n,t]: there is the problem

In[91]:= Clear[rv, M, n, x, k, eqm]

In[92]:= n = 2;

In[93]:= rv[t_, n_] := Table[x[k, t], {k, 1, n}]

In[94]:= rv[t, n]

Out[94]= {x[1, t], x[2, t]}

In[95]:= M[n_] := Table[Table[Random[], {n}], {n}]

In[96]:= MatrixForm[M[n]]

Out[96]//MatrixForm=
MatrixForm[{{0.4129438091150023, 0.19074665264305043}, 
{0.31296972444474486, 0.31543944991928186}}]

In[99]:= eqm = D[rv[t, n], t] == M[n] . rv[t, n]

Out[99]=
{Derivative[0, 1][x][1, t], Derivative[0, 1][x][2, t]} == 
{0.8654946712834201*x[1, t] + 0.6944612611502227*x[2, t],
    0.7720429142604556*x[1, t] + 0.016353185110213362*x[2, t]}

In[100]:= DSolve[eqm, rv[t,n], t]

 From In[100]:= DSolve::"nvld" : "The description of the equations 
appears to be ambiguous or \
invalid."

 From In[100]:= DSolve::"deqx" : "Supplied equations are not 
differential equations of the \
given functions."

 From In[100]:= DSolve::"deqx" : "Supplied equations are not 
differential equations of the \
given functions."

Out[100]= DSolve[{Derivative[0, 1][x][1, t], Derivative[0, 1][x][2, t]} 
== {0.8654946712834201*x[1, t] + 0.6944612611502227*x[2, t],
     0.7720429142604556*x[1, t] + 0.016353185110213362*x[2, t]}, {x[1, 
t], x[2, t]}, t]

 From In[111]:= DSolve::"nvld" : "The description of the equations 
appears to be ambiguous or \
invalid."

 From In[111]:= DSolve::"deqx" : "Supplied equations are not 
differential equations of the \
given functions."

 From In[111]:= DSolve::"deqx" : "Supplied equations are not 
differential equations of the \
given functions."


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