Now, let us consider a second-order linear ordinary differential equation, we may consider a general boundary value problem. But we discussed how to transform any general problem into self-adjoint form starting from (1.3). Therefore, we may consider the problem directly. Let
where L is the differential operator defined at (1.1).
We may assume the solution has the general form 
, where 
satisfies the boundary value problem
and 
is a solution of
The general solution for
is
where 
and 
are linearly independent solutions of the homogeneous differential equation.
The constant 
and 
are then determined by imposing the non-homogeneous boundary conditions.
Let us now consider the particular solution. We may consider the problem
as an example of the string deflection problem, 
would be the distribution function of the force density. And we first consider
the force concentrated at a point, say 
, and 
be a function of the deflection of string representing a unit of force
applying at the point
and satisfying the boundary conditions. Since the continuity of
, all other solutions are simply superposition of this function. Therefore,
the solution will be
If we formally apply the differential operator L to both sides of (3.5), and assuming commutativity of L with integration, we find that
If
is indeed a solution of the differential equation in (3.3),
the right hand side of (3.6) must be equal
to 
. This will lead to
In order to uniquely determine the function
, we must find conditions other than (3.7)
that contribute to its definition. Now, let us impose the homogeneous boundary
conditions in (3.3) on the solution, which
leads to
Since
can be any functions, these relations are possible only if
Consequently, we have shown that the Green's function we are seeking is a solution of the boundary value problem
where 
is fixed and 
.
Although this problem is quite similar to the problem described by (3.3),
the forcing function in (3.11) is a delta
function rather than an arbitrary function 
. This means that solving the problem for g will be somewhat simpler
than solving the corresponding problem for u. and once the Green's
function has been found for a particular operator L and set of boundary
conditions. It may be used for solving (3.3)
for any number of times where only the function
changes from problem to problem. It is the feature of the Green's function,
coupled with its physical interpretation, which makes it most useful in
applications.
The presence of the delta function in (3.11)
suggests that the behavior of
in the vicinity of
is somewhat peculiar. To investigate this behavior, we start with (3.7)
to obtain
Because the solution of a differential equation must be a continuous
function, the left-hand side of the above expression vanishes. And since
is arbitrary, we deduce that
which implies that
is continuous at
. Going one step further, we now wish to investigate the behavior of the
derivative of
. Here we formally integrate both sides of (3.7)
with respect to x from 
to 
, and find that
From the continuity of both q(x) and
at
, it follows that the integral on the left-hand side of this expression
is zero. Also, using the integral property of the delta function and the
fact that p(x) is continuous and non-zero on [a, b],
this last expression reduces to
where we have divided by 
. This result asserts that at
, the derivative of
has a jump discontinuity of magnitude 
.
Summarize all the above materials, we can well define that the Green's
function
associated with the boundary value problem
where L is self-adjoint operator as defined in (1.1).
is a function satisfying the following conditions
:
Based on the above four conditions, an explicit formula for the Green's function for one dimension can now be constructed.
