Circuit analysis is the process of finding all the currents and voltages in a network of connected components. We look at the basic elements used to build circuits, and find out what happens when elements are connected together into a circuit.
The basic elements of circuit design are components (resistors, capacitors, and inductors), and sources (voltage and current). We describe these elements as both ideal elements and real-world objects. We also develop precise terminology for how to talk about circuits and label voltages and currents.
If we limit ourselves to input signals shaped like sines or cosines, we can develop a whole AC analysis method to figure out what happens in circuits with changing signals. The methods are far simpler than solving a differential equation.