Homogeneous Solution

A homogeneous solution is a set of solutions to a differential equation where all parts of the solution satisfy the equation when set equal to zero. Essentially, it describes the behavior of a system without any external influences or inputs. Think of it as the natural response of a system, like how a pendulum swings back and forth when no outside force is acting on it. Homogeneous solutions help us understand the inherent properties of the system, serving as a foundation before considering external factors or forces.