Find the Differential Equation of an Analog Band Pass Filter I don't even know how to start. We're supposed to find the differential equations that links $V_{in}$, $V_{out}$, and $V_{io}$, as well as the values of $C_1$, $C_2$ and $R$ so the filter works in the audible spectrum. We're allowed to use Fourier transform, if that simplifies anything.



I thought of splitting the band pass filter as a high pass at first, and a low pass after.

That way I can relate $V_{io}$ with $V_{in}$, and $V_{out}$, with $V_{io}$, in two different equations.

Doing that... the results were:

$RC \frac{\mathrm{d} Vin(t)}{\mathrm{d} t} = Vio(t) + RC \frac{\mathrm{d} Vio(t)}{\mathrm{d} t}$

for the high pass filter

$Vio(t) = Vout(t) + RC \frac{\mathrm{d} Vout(t)}{\mathrm{d} t}$

for the low pass filter

Then, using..
$fc = \frac{1}{2 \pi R C}$

For the audible frequency, it'd be between $20$ hz- $20$ KHZ. So using that, I should be able to express both $C_1$ and $C_2$ as a function of $R$.

However, how do I actually determinate the value of $R$ required? or do I assume a set value?

If there's anything wrong, please let me know why. Thank you, have a nice day.
I'm afraid your first equation is wrong because it assumes a highpass filter without any load. However, there is a lowpass filter connected at its output and for this reason the current through the resistor leading to ground does not equal the current through $C_1$.

What you need to do is apply the complex voltage divider rule in the frequency domain, which will give you

$$\frac{V_{io}(\omega)}{V_{in}(\omega)}=\frac{Z(\omega)}{1/j\omega C_1+Z(\omega)}\tag{1}$$

where $Z(\omega)$ is the impedance of $R$ in parallel with the series connection of $R$ and $C_2$. I'm sure you can figure this out. The relation between $V_{out}$ and $V_{io}$ is simply

$$\frac{V_{out}(\omega)}{V_{io}(\omega)}=\frac{1/j\omega C_2}{1/j\omega C_2+R}=\frac{1}{1+j\omega RC_2}\tag{2}$$

The total response $V_{out}/V_{in}$ is simply obtained by multiplying Eq. (1) with Eq. (2). The time domain differential equations are obtained by multiplying Eqs (1) and (2) by the denominators of both sides and noting that a multiplication by $j\omega$ corresponds to differentiation.