When solving node/loop equations manually, it's generally necessary to
resort to phasor analysis with its underlying assumptions, or Laplace
transforms. The latter does have the capability of producing a time
response. But the solution requires finding the inverse transform, a
process similar to integration in that there's no single direct rule,
and often it's impossible to find a solution except for simple cases.

Computers can be programmed to solve complex problems numerically, using
fundamental time-domain current/voltage relationships (such as the
relationship V(t) = L di/dt for an inductor, or even more generally,
V(t) = L di/dt + I dl/dt for a time-varying inductance). This is
basically what SPICE does, and it's able to easily solve problems which
are simply not possible to do manually, either because of the enormous
time that would be required, or the impossibility of finding a reverse
Laplace transform -- or its equivalent, the solution to a high-order
differential equation if Laplace transforms aren't used. A google search
on 'SPICE "time step" equations' brought a number of hits. I'm sure you
can find an adequate explanation of the inner workings of SPICE among them.

Roy Lewallen, W7EL

Steve Nosko wrote:
. . .
Correct me if I am wrong (like I need to say this here, eh?)
I believe the underlying basis is the collection of loop / node equations
used (by Engineers) to model circuits. We know the behavior of resistors,
inductors and capacitors and have mathematical models for them. To this we
add the active devices, etc. and develop an "engine" which does all the
calculations for us. [[we used to do them by hand/slide rule -- yes, I am
included in this we]]. These loop and node equations provide us with a
mathematical model of the behavior of electronic circuits. If done
carefully, this is a general purpose model which applies to all the
situations for which our component models are valid.
Some time later there were bare engines into which we had to type the part
values and node numbers (the sane things you can see in printouts from
Spice). As computers got more powerful, schematic entry was developed. I
believe these programs to be very useful, but as with any model or
simulation, it is best to understand the limitations.
Thre is an alternate method. It is also possible to derive equations for
each type of situation and use these calculations each time you need to
solve that type of problem. I am sure you are familiar with the equations
for things such as parallel capacitors and resonance and so forth. These
are specific solutions of the properties of components in those specific
circuits.
. . .