Express ODEs in matrix form

Question

Is it possible to use Mathematica to express 2 simultaneous ODEs in matrix form? My objective is to then take the matrices and solve the ODE elsewhere.

I have formulated a simple system of ODEs that represent some electrical system:

eqn = {Ih == C1(D[Va[t],t]) +(Va[t] - Vb[t])/R3,
       (Va[t] - Vb[t])/R2 == C2(D[Vb[t],t]) + Vb[t]/R3}

How can I then use Mathematica to rearrange these equations in a matrix form? I want to get to the point where I can solve the system for the next time increment using a matrix inversion.

Answer 1

eqn = {Ih == C1 (D[Va[t], t]) + (Va[t] - Vb[t])/R3, 
       (Va[t] - Vb[t])/R2 == C2 (D[Vb[t], t]) + Vb[t]/R3} /. {D[Va[t], t] -> Va'[t], 
        D[Vb[t], t] -> Vb'[t]};
sort = First@Solve[eqn, {Va'[t], Vb'[t]}] /. Rule -> Equal

m = -CoefficientArrays[sort, {Va[t], Vb[t], Ih}][[2]] // Normal;
A = m[[All, 1 ;; 2]];
b = m[[All, 3]];
{A // MatrixForm, b // MatrixForm}

Supplement

There is another way:

eqn = {Ih[t] == C1 (D[Va[t], t]) + (Va[t] - Vb[t])/R3, 
(Va[t] - Vb[t])/R2 == C2 (D[Vb[t], t]) + Vb[t]/R3} /. {D[Va[t], t] -> Va'[t], 
        D[Vb[t], t] -> Vb'[t]};  
ss = StateSpaceModel[eqn, {Va[t], Vb[t]}, {Ih[t]}, Vb[t], t]

{aa, bb, cc, dd} = Normal@ss;
{aa // MatrixForm, bb // MatrixForm}