nonhomogeneous equation

Picture a child on a swing. If you let go and just watch, the swing settles into its own natural motion — that is the homogeneous (unforced) story. But if someone keeps pushing the swing in rhythm, you have added an outside influence: a forcing. The nonhomogeneous equation is the model with that outside push written in.

Concretely, a second-order linear equation has the form a(x) y'' + b(x) y' + c(x) y = g(x). When the right-hand side g(x) is exactly zero, the equation is called homogeneous; when g(x) is anything other than zero, it is nonhomogeneous. That g(x) is the forcing function — the part of the equation that does not depend on y or its derivatives at all, but is supplied from outside. After dividing through by the leading coefficient, the standard form is y'' + p(x) y' + q(x) y = g(x).

The word 'nonhomogeneous' simply marks 'there is a nonzero right-hand side.' This matters because the whole solution strategy splits in two: first understand the unforced system (its homogeneous solutions), then find one solution that copes with the forcing. Almost every applied model — a circuit driven by a voltage source, a mass pushed by an external force, a beam under a load — is nonhomogeneous, so this is the equation you usually actually face.