homogeneous and inhomogeneous PDEs
There is a simple yes/no question you can ask of any linear PDE that turns out to matter enormously: is anything driving it from outside? An equation is homogeneous if every term involves the unknown u (or its derivatives), so that u = 0 is automatically a solution — nothing is being added in. It is inhomogeneous (or nonhomogeneous) if there is an extra term that does not contain u at all: a source, a forcing, a feeding-in that keeps u from simply being zero.
Written as L[u] = g for a linear operator L, the equation is homogeneous when g = 0 and inhomogeneous when g is some given nonzero function of the independent variables. Laplace's equation Laplacian u = 0 is homogeneous — empty space, no charges; Poisson's equation Laplacian u = f is the inhomogeneous version, where f is a distribution of charge or mass that sources the potential. The plain heat equation u_t = k u_xx is homogeneous (heat just redistributes itself); add a heater and you get u_t = k u_xx + Q(x, t), inhomogeneous, with Q pumping energy in.
The distinction is the backbone of how linear problems are solved. The general solution of an inhomogeneous linear PDE is one particular solution (any single solution of the full equation) plus the general solution of the associated homogeneous equation — exactly as for linear algebra and linear ODEs. So you split the work: solve the homogeneous equation to capture the freedom (this is where superposition and eigenfunction expansions live), then find any one particular response to the source. Methods like Duhamel's principle and Green's functions are systematic machines for building that particular solution out of the source term.
Homogeneous / inhomogeneous pairs: Laplacian u = 0 vs Laplacian u = f (Laplace vs Poisson); u_t = k u_xx vs u_t = k u_xx + Q(x, t); u_tt = c^2 u_xx vs u_tt = c^2 u_xx + F(x, t) (free vibration vs a driven string). The added term on the right is the source.
Same operator on the left; a nonzero right-hand side turns a homogeneous equation into an inhomogeneous one.
Beware two different uses of 'homogeneous' in PDE: here it means 'no source term'. The same word is also used for 'homogeneous boundary conditions' (boundary data set to zero) — a related but separate notion. Context tells you which is meant; both can be present or absent independently.