For any coefficient matrix A(t), the equation
y’ = A*y + f’ – A*f
has y=f as a solution. A perfect emulator. But as I showed above, the error propagation is given by the homogeneous part y’ = A*y. And that could be anything at all, depending on choice of A. Sharing a common solution does not mean that two equations share error propagation. So it’s not OK.