The exercise involves solving a problem related to an RC dipole (resistor-capacitor circuit) described in Figure 1 and Figure 2. Below is a breakdown of the questions and the general steps to approach the solutions:

1. Show that the voltages $U_1(t)$ and $U_2(t)$ are related by the equation $U_2(t) = \frac{U_1(t)}{3}$:

   • From the circuit in Figure 1, the total resistance is composed of $R$ and $r$, where $R = 3r$.
   • Since $U_1(t)$ and $U_2(t)$ are the voltages across resistors, the voltage divider rule applies. The ratio between the resistors dictates the relationship between the voltages, thus confirming the equation.

2. Establish the differential equation for $U_1(t)$ at the capacitor $C_1$:

   • This is a first-order RC circuit, so the voltage across a capacitor is governed by the differential equation: $U_1(t) = U_0 e^{-\frac{t}{\tau}}$ where ( \tau = R \cd