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3 Alternative Construction Procedure As can be found in classical textbooks on differential equations, the notion of a Green’s function is introduced for a boundary-value problem where both the governing differential equation and the boundary conditions are homogeneous. Such settings are referred to as homogeneous boundary-value problems. In this section, we will turn our attention to inhomogeneous linear differential equations subject to homogeneous boundary conditions. 34 Chapter 1 Green’s Functions for ODE In this section, we will state an prove an important theorem.

This implies that the conditions for existence and uniqueness of its Green’s function are satisfied. 81)) do not show up in this case. 99) This formula transforms into a single integral with its kernel expressed in two pieces. 3 Alternative Construction Procedure Note that the kernel in the first of the above two integrals is valid for x s, because x represents its upper limit of integration, whereas the kernel in the second integral, by similar reasoning, is valid for x Ä s. a s/; for x Ä s, x/; for s Ä x.

70). a; b/. x/; for s Ä x Ä b. x/ satisfy the first and the second of those boundary conditions, respectively. 72) already meets properties 1 and 4 of the definition of Green’s function. s/. s/. x/; for s Ä x Ä b. Hence, the theorem has been proven. x; s/ of a self-adjoint boundary-value problem is invariant under exchange of the observation point x and the source point s. x; s/, we obtain the other branch. The symmetry of Green’s functions, the analysis of which we completed, has important applications in various applied sciences.

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