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2. Two-Point Boundary Value Problems
42
Example 2.2 Consider the problem (2.1) with f(x)= x. Again, from (2.7)
we get
x
1

1
(x − y)ydy =
x(1 − x ).
u(x)= x
(1 − y)ydy −
2
6
0
0

Further examples of how to compute the exact solution formulas for two-
point boundary value problems are given in the exercises. In Project 2.1
we will also see how the exact representation of the solution can be used
to derive numerical approximations when the integrals involved cannot be
evaluated analytically.
2.1.1 Green’s Function
The unique solution of (2.1) can be represented in a very compact way by
introducing an auxiliary function: the Green’s function.
Introduce the function

y(1 − x)if 0 ≤ y ≤ x,
G(x, y)= (2.8)
x(1 − y)if x ≤ y ≤ 1.
It follows that the representation (2.7) can be written simply as

1
u(x)= G(x, y)f(y) dy. (2.9)
0
The function G is called the Green’s function for the boundary value prob-
lem (2.1), and it has the following properties:
• G is continuous,
• G is symmetric in the sense that G(x, y)= G(y, x),
• G(0,y)= G(1,y)= G(x, 0) = G(x, 1)=0,
• G is a piecewise linear function of x for fixed y, and vice versa,
• G(x, y) ≥ 0 for all x, y ∈ [0, 1].
These properties follow directly from (2.8). The function is plotted in
Fig. 2.1.
Of course, the representation (2.9) is only a reformulation of (2.7). How-
ever, the representation (2.9) is very convenient when we want to derive
various properties of the solution u.