To obtain the mathematical model for the circuits provided in the image, let's analyze each one in detail.

Circuit on the Left: Inverting Operational Amplifier with Resistor $R$ and Capacitor $C$

1. Configuration: This is an inverting amplifier circuit with a resistor $R$ connected to the input and a capacitor $C$ in the feedback loop.

2. Kirchhoff's Current Law (KCL): At the inverting input node, the sum of currents should be zero: $\frac{V_{in} - V_{-}}{R} + C \frac{d(V_{-} - V_o)}{dt} = 0$ Since the inverting input is connected to a virtual ground (ideal op-amp assumption, where $V_{-} = 0$), this simplifies to: $\frac{V_{in}}{R} + C \frac{d(V_o)}{dt} = 0$

3. Differential Equation: $V_{in} = -RC \frac{dV_o}{dt}$ This equation describes the relationship between the input voltage $V_{in}$ and the output voltage $V_o$.

Circuit on the Right: Capacitor and Resistor with Operational Amplifier

1. Configuration: In this configuration, the input voltage $V_{in}$ is passed through a capacitor $C$, followed by a resistor $R$ at the output of the operational amplifier.

2. Low-Pass Filter Behavior: The capacitor $C$ blocks any DC component, allowing only AC signals to pass. The resistor $R$ forms a low-pass filter with the capacitor.

3. Voltage Relationship: At the output node, considering the capacitor $C$ and resistor $R$, the transfer function for a simple low-pass filter is: $\frac{V_{out}}{V_{in}} = \frac{1}{1 + j \omega RC}$ This transfer function shows that at higher frequencies, the output will be attenuated, and at lower frequencies, the output will approximate the input voltage.

In summary:

1. First Circuit (Inverting Op-Amp with Capacitor): The equation is $V_{in} = -RC \frac{dV_o}{dt}$.
2. Second Circuit (Low-Pass Filter with Capacitor and Resistor): The transfer function is $\frac{V_{out}}{V_{in}} = \frac{1}{1 + j \omega RC}$.

Would you like more details on any of the components or their functions?

Here are some related questions you might explore:

1. How does the frequency response of the second circuit change with $R$ and $C$?
2. What would happen if the positions of the capacitor and resistor were swapped in either circuit?
3. How do the time-domain and frequency-domain representations relate for each circuit?
4. Can the first circuit be considered a high-pass filter? Why or why not?
5. What are the practical applications of each of these circuits in signal processing?

Tip: For both circuits, the capacitor's role is crucial in controlling how the circuit behaves with changing frequencies, especially in the context of signal filtering.