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I have two differential equations that are connected by an equation,

$L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$

$L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$

$I_1+I_2=I$

where $L_1, L_2, C_1, C_2$ are constants. As one might see, the system that is described by the differential equations consists of two LC circuits that are connected in parallel.

How do I write this as one equation without $I_1$ and $I_2$?

I've tried several things but I think I'm missing something, every time I end up with two equations that I can't seem to substitute in each other.

Thank you.


This is what I've done.

The aim is to model the behavior of a circuit by V and I with a differential equation, the circuit consists of two LC circuits in parallel, this results in:

Kirchhoffs laws,

$I = I_1 + I_2$

$V = V_{L_1} + V_{C_1}$

$V = V_{L_2} + V_{C_2}$

Capacitor and inductor equations

$I_1 = C_1 \frac{dV_{C_1}}{dt}$

$I_2 = C_2 \frac{dV_{C_2}}{dt}$

$V_{L_1} = L_1 \frac{dI_1}{dt}$

$V_{L_2} = L_2 \frac{dI_2}{dt}$

System

Differentiating the voltage laws and substituting for the capacitor and inductor equations results in $\frac{dV}{dt} = \frac{dV_{L_1}}{dt} + \frac{dV_{C_1}}{dt} = L_1 \frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1$

$\frac{dV}{dt} = \frac{dV_{L_2}}{dt} + \frac{dV_{C_2}}{dt} = L_2 \frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2$

Only thing that is left is to substitute with the current law, substituting with $I_2 = I - I_1$

$\frac{dV}{dt} = L_1 \frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1$

$\frac{dV}{dt} = L_2 \frac{d^2I}{dt^2} - L_2 \frac{d^2I_1}{dt^2} + \frac{1}{C_2}I - \frac{1}{C_2}I_1$

This is where I get stuck, how do I combine these two?

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  • $\begingroup$ IMHO you cannot. The last equation is not restriction $I$ just derived from $i_1$ and $I_2$. What is the problem to solve them for $I_1$ and $I_2$ and make sum at the end? $\endgroup$ – Alexander Vigodner Sep 18 '14 at 22:18
  • $\begingroup$ How would that be possible, because both equations depend on $\frac{dV}{dt}$? $\endgroup$ – Donzos Sep 19 '14 at 7:31
  • $\begingroup$ How $V$ is related to $I$? You did not specify. If not I still don't understand you problem. Is $V$ known input? $\endgroup$ – Alexander Vigodner Sep 19 '14 at 12:17
  • $\begingroup$ I have added an image to make the problem more clear, $V$ is not directly related to $I$ I think. $V$ is not known, the idea is to describe the behavior of the system, this is already achieved with just the Kirchhoffs laws and the capacitor and inductor equations, I want to describe the behavior without the latent variables, thus an expression containing just $V$ and $I$ (and their derivatives). $\endgroup$ – Donzos Sep 19 '14 at 12:47
  • $\begingroup$ I repeat my opinion. I think you cannot. You have to solve it for $I_1$ and $I_2$ (assuming that $V$ is a known input, you did not answer this question and I forgot pretty much from this theory so I am looking only on equations without thinking about physics) $\endgroup$ – Alexander Vigodner Sep 19 '14 at 15:06

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