Need some help on Circuit modeling(1 sample)

I'm learning circuit modeling recently, and got stuck by a simply serial circuit sample(shown in the attachment), here're equations I wrote for the sample, but I have no idea what initial conditions and algorithm I can use to solve it, could any one give me a hand or some tips?

[LATEX]i_{21}+\int_0^t \frac{V_3-V_1}{L}d \tau =0[/LATEX]


[LATEX]i_{12}+\frac{GND-V_2}{R}=0[/LATEX]

[LATEX]\int_0^t \frac{V_1-V_3}{L} d\tau +C \cdot (\frac{dGND}{dt}-\frac{dV_3}{dt})=0[/LATEX]

[LATEX]C \cdot \frac{dV_3}{dt}+\frac{V_2-GND}{R}=0[/LATEX]


where GND=0 V is constant, I see that [LATEX]i_{12}=- i_{21}[/LATEX] can be used to reduce the equations, but then the remaining equations are 2nd order diff equations, how do computers solve this?

[LATEX]\int_0^t \frac{V_2-V_1}{L}d\tau = -\frac{-V_2}{R}[/LATEX]

[LATEX]\int_0^t \frac{V_1-V_3}{L}d\tau + C \cdot -\frac{dV_3}{dt}=0[/LATEX]

[LATEX]C \cdot \frac{dV_3}{dt}+\frac{V_2}{R}=0[/LATEX]

Forgive me, but what are you solving for?

As to values, I'd think you could arbitrarily pick any value you like for your inductor, the cap, the resistor, time and (although I don't see any variables for them) the frequency and voltage level outputs of the sig gen.

Plug the #s into the integrals and "wail", as we used to say.

Transform everything into the Laplace domain. So where you see an integral, replace the integral with 1/s. Likewise, replace all the differentials with s. This gives you a set of equations without any of the integrals. The currents should drop out from there

I wanna know, if a.c. src u=sin(wt) and other parameters C,L,R are all given , then how can I figure out a numerical solution for the current i (by algorithm)?

Terribly Sorry for not writing a clear question.

simonbramble said:

Transform everything into the Laplace domain. So where you see an integral, replace the integral with 1/s. Likewise, replace all the differentials with s. This gives you a set of equations without any of the integrals. The currents should drop out from there

Click to expand...

Excuse me, what does "s" stands for here? I supposed that the equations should be first taken derivatives then it becomes a 2nd order diff equations set, can you introduce some tutorials about your method?

Computers don't generally solve the closed form of the equations, they use iterative algorithms to solve such problems. Here's a reference for those typical of Spice type simulators.

crutschow said:

Computers don't generally solve the closed form of the equations, they use iterative algorithms to solve such problems. Here's a reference for those typical of Spice type simulators.

Click to expand...

Thanks a lot!