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
$I_1 = C_1 \frac{dV_{C_1}}{dt}$
$I_2 = C_2 \frac{dV_{C_2}}{dt}$
$V_{L_1} = L_1 \frac{dI_1}{dt}$
$V_{L_2} = L_2 \frac{dI_2}{dt}$
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?