Find the Differential Equation of an Analog Band Pass Filter
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.