What is the physical interpretation of homogeneous? In ode's and pde's we pay great attention as to whether the equations are homogeneous or nonhomogeneous. I remember learning in my first ODE class that for the general linear ode
$$a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1(x)\frac{dy}{dx}+a_0(x)y=g(x),$$
that $g(x)$ takes on some very important physical meanings in engineering problems but I can't remember what they are. And in general, could someone provide an interpretation of the physical meaning of g(x) in odes and in pdes? If your examples are only from famous and specific equations then that's welcome too.
 A: $g$ is interpreted as the source/forcing term. A couple of examples

Oscillator

If you attach a mass $m$ to a spring of elastic constant $k$ and drive the system with a external force $f_{\rm ext}$ then Newton's second law applied to the mass gives you
$$
-kx + f_{\rm ext} = m \frac{{\rm d}^2 x}{{\rm d}t^2}
$$
where $x$ labels the position of $m$. Rearranging this equation you end up with
$$
\frac{{\rm d}^2x}{{\rm d}t^2} + \omega^2 x  = g
$$
where $g = f_{\rm ext}/m$. From this example you can immediately tell why the $g$ term is known to be a "forcing term"

Gravity

If $\Phi$ denotes the gravitational potential then Poisson's equation 
$$
\nabla^2 \Phi = 4\pi G\rho
$$
expresses how the mass density $\rho$ affects the field $\Phi$. In fact, it states that the source of $\Phi$ is $\rho$

Maxwell Equations

In the same spirit, the equations
$$
\partial _\alpha F^{\alpha\beta} = \mu_0 J^\beta
$$
tell you that the source of electromagnetic fields are charges/currents
A: It is often a source term providing an action to the system being modeled. For example, in the heat equation
$$u_t - \Delta u = f,$$
$f$ is a heat source density, representing the rate at which heat is being added/removed to/from the system at each point in space-time. For the wave equation
$$u_{tt} - \Delta u = f,$$
$f$ represents a transversal force density applied to the vibrating string or membrane (it's really just Newton's law). 
