the forcing function

Think of the equation as a system being acted on from outside. The forcing function is that outside action: the term on the right-hand side that does not involve the unknown y or its derivatives at all. In a mechanical model it is the externally applied force; in a circuit it is the applied voltage; in a thermal model it is the heat being supplied.

In the standard form y'' + p(x) y' + q(x) y = g(x), the forcing function is g(x). Everything on the left describes the system's own dynamics; g(x) on the right is the input. When g(x) = 0 the equation is homogeneous and the system simply runs on its own (free response); when g(x) is nonzero the system is being driven, and the shape of g(x) dictates which solution method is convenient — a polynomial, exponential, or sinusoidal g invites undetermined coefficients, while a more exotic g sends you to variation of parameters.

The forcing function is the bridge between a tidy classroom equation and a real input signal. Engineers often call it the input and the resulting y the output, and the central question becomes how a given input produces its output. One thing to keep straight: g is a known, prescribed function of the independent variable only — it must not contain y, y', or y''; if it does, you no longer have a simple forced linear equation in this sense.