Implicit ODE’s, Rosenbrock Formulation & the Josephson’s Junction:

ODE’s of the form

are referred to as implicit ODE’s. While a matrix inversion can be used to convert these to an explicit ODE, the form of the Rosenbrock integration scheme can be modified to directly solve for an implicit 1^st order set of ODE’s.

The complete integration can be performed using the same procedures described earlier.

Getting back to numerically solving the Josephson’s Junction equations, which is an implicit ODE, the above formulation can also be used as an alternative and avoid the matrix inversion mentioned earlier.