Throughout the previous discussion we have neglected the presence of the
time-dependent term 
in equation
(
). If we are to solve a time-dependent problem it is
necessary to include this term in the residual formed in equation
(). This will give rise to an additional term in the matrix
equation, traditionally called the capacitance matrix, 
.
Here 
is given by
We can develop an implicit, numerically stable, backward-difference scheme by using the following approximation, where n indicates the time step.
Substituting equation () into () and
suppressing subscripts and summation we obtain an expression relating
the solution at one time step to the solution at an ensuing time step.
Interestingly this form of the equation is identical to the form of the steady state solution where the stiffness and load vectors are modified by the capacitance matrix.
