26 Chapter 0 Ordinary Differential Equations
0.3 Boundary Value Problems
Aboundary value problemin one dimension is an ordinary differential equa-
tion together with conditions involving values of the solution and/or its deriv-
atives at two or more points. The number of conditions imposed is equal to
the order of the differential equation. Usually, boundary value problems of any
physical relevance have these characteristics: (1) The conditions are imposed
at two different points; (2) the solution is of interest only between those two
points; and (3) the independent variable is a space variable, which we shall
represent asx. In addition, we are primarily concerned with cases where the
differential equation is linear and of second order. However, problems in elas-
ticity often involve fourth-order equations.
In contrast to initial value problems, even the most innocent looking
boundary value problem may have exactly one solution, no solution, or an
infinite number of solutions. Exercise 1 illustrates these cases.
When the differential equation in a boundary value problem has a known
general solution, we use the two boundary conditions to supply two equations
that are to be satisfied by the two constants in the general solution. If the dif-
ferential equation is linear, these are two linear equations and can be easily
solved, if there is a solution.
In the rest of this section we examine some physical examples that are natu-
rally associated with boundary value problems.
Example: Hanging Cable.
First we consider the problem of finding the shape of a cable that is fastened
at each end and carries a distributed load. The cables of a suspension bridge
provide an important example. Letu(x)denote the position of the centerline
of the cable, measured upward from thex-axis, which we assume to be hori-
zontal. (See Fig. 3.) Our objective is to find the functionu(x).
The shape of the cable is determined by the forces acting on it. In our analy-
sis, we consider the forces that hold a small segment of the cable in place. (See
Fig. 4.) The key assumption is that the cable is perfectly flexible. This means
that force inside the cable is always a tension and that its direction at every
point is the direction tangent to the centerline.
Figure 3 The hanging cable.