Laplace Transform.

[Dave_p]

It's been a while, so pardon the dust in my brain.

Can we apply a Laplace transform to a differential equation (circuit equation) when there isn't a closed-form solution?

For example, if the circuit has a feedback line, there isn't a closed form...correct...and (presumably) we then couldn't use the Laplace transform to find the transfer equation...(???)

Meme:

[mystran]

It's been a while, so pardon the dust in my brain.

Can we apply a Laplace transform to a differential equation (circuit equation) when there isn't a closed-form solution?

For example, if the circuit has a feedback line, there isn't a closed form...correct...and (presumably) we then couldn't use the Laplace transform to find the transfer equation...(???)

What do you mean by "closed form"?

If it's a linear system, then write in space-space form and you can get the transfer function as:
C*((Is-A)^-1)*B+D for continuous-time (state-variables are integrators) or
C*((Iz-A)^-1)*B+D for discrete time (state-variables are unit delays).

If it's a circuit, then you can also observe that in "modified nodal analysis" the "MNA matrix" is really the same thing as the (Is-A) matrix in the state-space formulation, except some dimensions might not have the "s" on the diagonal. No big deal, that turns it into a DAE instead of ODE, but in practice it usually solves just the same.

[Dave_p]

Ah, yup, that answered the question. Got it. Laplace transform on the matrix, and we still can get some solution to the differential equation...so all the variables can be reduced down to f(x) = y...

And I'm back on to my random existential problems and thoughts...'I wonder how I can apply a laplace transform to my life to determine the frequency of my blunders.'