How to get companion system in Mupad

Hi,

I would like ask how to get the companion system of ODE in Mupad. I have system of 2 ODE's first is 2'nd order second is 1'st order.

They look like this (mupad syntax)

Lt :=[ L*diff(i_1(t),t$2)+(i_1(t)-i_2(t))/C, R*diff(i_1(t),t$1)+(i_2(t)-i_1(t))/C] ;

I know how to get the companion systems of one equation, using this function. ode::companionSystem(Equa, Var) ; If I apply this function to my system ODE's it does not function.

Could somebody give me any advice? Thanks Jan

Answer 1

[Andrew Newell]

You could eliminate the variable i2 by solving for it in the second equation and substituting in the first. Then you have a second order ODE for i1 alone, and you can find the companion system for that.

Answer 2

[Jan Svanda]

Thank you for the answer.

It is true, but a would to get the companion system without eliminating. I solved it "manualy"(you cen see it below) and I would like to obtain the same solution programaticaly. Is there any smart solution :-) ?

diff(x_1(t), t)=x_2(t);
diff(x_2(t), t)=1/(L*C)*(-x_1(t)+i_2(t));
diff(x_2(t), t)=1/(R*C)*(x_1(t)-i_2(t));

A:=matrix([[0,1,0],[-1/(L*C),0,1/(L*C)],[1/(R*C),0,-1/(R*C)]])

Answer 3

[Christopher Creutzig]

I am not sure what the companion system of a system of differential equations would be, the definition does not, in my opinion, canonically generalize. However, if what you are looking for is what I would call a phase space representation (which your answer seems to suggest), you may be interested in numeric::ode2vectorfield, as in:

Lt :=[i_1'(t) = L*diff(i_1(t),t$2)+(i_1(t)-i_2(t))/C,
      i_2'(t) = R*diff(i_1(t),t$1)+(i_2(t)-i_1(t))/C]:
fields := [i_1(t), i_1'(t), i_2(t)]:
fn := numeric::ode2vectorfield(
  Lt . map(fields, f -> (f | t=0) = dummy), fields)[1]:
zip(map(map(fields, diff, t), rewrite, D), fn(t, i), `=`)

    [i_1'(t) = i[2],
     i_1''(t) = (C*i[2] - 1.0*i[1] + i[3])/(C*L),
     i_2'(t) = (i[3] - 1.0*i[1] + C*R*i[2])/C]

This has some drawbacks and may not be the final solution for you:

• numeric::ode2vectorfield will call float on its input, so you may need to work with rationalize if you want to preserve constants like ?, and use numeric::rationalize on the output, too.
• It does not create the list fields for you, but will complain if that list does not work for the input you have.
• numeric::ode2vectorfield will not do symbolic differentiation etc. in its computation, but instead will simply refuse to work on inputs that are not quasi-linear.
• I may have completely misinterpreted your question and this may not at all be what you are looking for in the first place. Then again, it may be a step that can become useful in a larger program. :-)