This requires differential elimination, which in turn requires prolongation. The latter simply involves taking derivatives of the existing expressions and setting them to zero. The former is a matter of algebraic elimination of variables.
Rule of thumb: elimination of n
variables requires n+1
equations. We have two equations initially, and voltageC0
appears up to order 2. Noticing that each new derivative gives two new equations and adds one to the highest derivative of the term being eliminated, we decide we need to take two derivatives (so we get six equations and voltageC0
might appear in derivatives to order zero through 4, that is, five terms to eliminate).
With these considerations, the code below does the prolongation (twice), and obtains the variables to eliminate and the ones to keep.
odes = {-voltageC0[t] - 2 voltageC0''[t] +
2 voltageC1''[t], -voltageC1[t] - 2 voltageC1''[t] +
2 voltageC0''[t]};
prolonged = Join[odes, D[odes, t], D[odes, {t, 2}]]
allvars = Variables[prolonged];
elims = Select[allvars, ! FreeQ[#, voltageC1] &]
keepvars = Complement[allvars, elims]
(* Out[102]= {-voltageC0[t] - 2*Derivative[2][voltageC0][t] +
2*Derivative[2][voltageC1][t],
-voltageC1[t] + 2*Derivative[2][voltageC0][t] -
2*Derivative[2][voltageC1][t],
-Derivative[1][voltageC0][t] - 2*Derivative[3][voltageC0][t] +
2*Derivative[3][voltageC1][t], -Derivative[1][voltageC1][t] +
2*Derivative[3][voltageC0][t] - 2*Derivative[3][voltageC1][t],
-Derivative[2][voltageC0][t] - 2*Derivative[4][voltageC0][t] +
2*Derivative[4][voltageC1][t], -Derivative[2][voltageC1][t] +
2*Derivative[4][voltageC0][t] - 2*Derivative[4][voltageC1][t]}
Out[104]= {voltageC1[t], Derivative[1][voltageC1][t],
Derivative[2][voltageC1][t],
Derivative[3][voltageC1][t], Derivative[4][voltageC1][t]}
Out[105]= {voltageC0[t], Derivative[1][voltageC0][t],
Derivative[2][voltageC0][t],
Derivative[3][voltageC0][t], Derivative[4][voltageC0][t]} *)
Now Eliminate
or GroebnerBasis
can be used to do the elimination step.
GroebnerBasis[prolonged, keepvars, elims]
(* Out[106]= {voltageC0[t] + 4*Derivative[2][voltageC0][t]} *)