the Green's function
/ GREEN's /
Imagine poking a stretched drumhead at a single point and recording the dent that results everywhere on the drum. Now any general load on the drumhead — pressure spread out over the whole surface — can be built by adding up the dents from a poke at each point, scaled by how hard you press there. The Green's function is exactly that record of the response to a single, concentrated unit poke; once you have it, you can solve the problem for any load by superposition.
Precisely, for a linear boundary value problem L[y] = f(x) with given homogeneous boundary conditions, the Green's function G(x, s) is the solution when the forcing f is a unit impulse, a Dirac delta delta(x - s) concentrated at the point s. The solution for a general forcing is then the integral y(x) = integral from a to b of G(x, s) f(s) ds — you weight the impulse-response by the actual load and sum (integrate) over all source points s. The function G satisfies the equation everywhere except at x = s, meets the boundary conditions, is continuous at s, and has a built-in jump in its slope there of size set by the delta. In this sense G is the inverse of the operator L: where L turns y into f, integrating against G turns f back into y.
The Green's function is powerful because it packages the entire solution operator into one object, computed once and reused for every right-hand side. It cleanly separates the equation and boundary conditions (which determine G) from the particular forcing (which just gets integrated in). It connects beautifully to eigenfunctions, since G can be written as a sum over modes y_n(x) y_n(s) / lambda_n, and it generalises directly to the partial differential equations of electrostatics, heat flow, and quantum scattering, where it is one of the central tools of mathematical physics.
For -y'' = f(x) on [0, 1] with y(0) = y(1) = 0, the Green's function is G(x, s) = x(1 - s) for x <= s, and s(1 - x) for x >= s. Then y(x) = integral from 0 to 1 of G(x, s) f(s) ds solves the BVP for any f. Note G is continuous at x = s but its slope jumps there — the signature of the unit impulse.
Build G once from the impulse response; then every forcing f is solved by a single integral against it.
A Green's function exists exactly when the corresponding homogeneous BVP has only the trivial solution — that is, when 0 is not an eigenvalue. If lambda = 0 is an eigenvalue, the operator is not invertible and no ordinary Green's function exists; you need a modified one and a solvability condition on f.