# Differential Equation From Impedance

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You're walking in the woods. There's no one around, and your phone is dead. Out of the corner of your eye you spot the following circuit:

You walk closer and get a better look at it:

Below it, written in blood^{1} are the instructions that you need to generate a differential equation for the voltage across capacitor C_2. Or more specifically you need to generate the characteristic equation of that differential equation's homogenous expression. If you do this, no harm will come to you.

"Hold on," you say, concern bubbling up in your speech. "There's more than two non-combinable energy storage components. That means..." You pause and breathe in and out before continuing. "That means that the differential equation will most likely be more than second order."

Thankfully 6.002 introduced the idea of impedance in lecture and this will help you figure it out. Generate a transfer function for the circuit above in the form of:

While deriving this expression do not simplify or combine your j\omega terms. Keep them original (so if you have j\omega \cdot j\omega, leave it as (j\omega)^2).

Because our sinuosidal signals can be reprsented (in time) as:

we can notice that if we asked for the derivative with respect to time, you'd get:

(remember \tilde{V} is just the phasor form of the voltage's amplitude and phase at the frequency of interest...it has not time dependence)

The second derivative would be:

and so on.

Using this fact return to your transfer function that you derived above, and using cross-multiplication, extract the homogenous form of this system's differential equation and express it's characteristic equation (using the variable `s`

to represent to represent the roots). For example, if the it is a first order differential equation you'd have something like `s+R1*C`

, etc.

Of course use standard notation that we've had previously, like C_1 is `C1`

.

**Footnotes**