Differential equations prove exceptional at modeling electrical circuits. In fact the very simple circuit, which is fundamental to larger circuit building, and three of the most fundamental electrical objects, a resistor, a capacitor, and inductor, can be modeled by a constant coefficient, linear, second order differential equation. We consider a RLC circuit show in the Modeling Scenario.
The EMF E(t) represents an Electromotive Force generated by an excess of electrons on one side of a barrier (the Switch) and a paucity of electrons on the other side of the barrier. When the switch is thrown the electrons in the excess area (say to the left of the circle marked EMF) seek to take the path of least resistance to get to the location of the paucity of electrons (to the right of the circle marked EMF).
Thus we say there is a potential awaiting the switch to be thrown and that potential (just like water held high and then released to run down a descending track) causes electrons to flow clockwise through the capacitance (C), through the resistance (R), and finally through the inductance (L) until these electrons are ``home" to the region of paucity of electrons.