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]
[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]
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