A Green's function describes the response of a linear differential operator to a point source. If $$Ly=f(x),$$ define $G(x,x')$ by $$LG(\cdot,x')=\delta(x-x')$$ with the same boundary conditions required of the physical solution.

Linearity then gives $$y(x)=\int G(x,x')f(x'),dx'.$$ The solution to an arbitrary source is built by superposing responses to infinitesimal point sources.

For the simple operator $$L=-\frac{d^2}{dx^2}$$ on an interval with fixed boundary conditions, $G$ is piecewise linear in $x$ away from $x=x'$ because $G''=0$ there. Its derivative has a jump at $x'$ chosen so that integrating the differential equation across the source produces the delta function.

The same idea appears in many forms. Electrostatic potential is the convolution of charge density with the Green's function of Poisson's equation. Linear time-invariant systems use an impulse response, which is a time-domain Green's function. Quantum propagators describe amplitudes for evolution between spacetime points.

A Green's function depends on both the operator and the boundary or causal conditions. Changing from free space to a conducting enclosure, for example, changes the Green's function even when the local differential equation is unchanged.

Green's functions convert solving a differential equation repeatedly for many sources into finding one reusable response kernel.