32 Chapter 0 Ordinary Differential Equations
Figure 7 Column carrying loadP.Figure 8 Section of column showing forces and moments.been of second order, causing the appearance of two arbitrary constants, which
are to be determined by the boundary conditions.
The next example is somewhat different in spirit from the others. Instead of
just finding the solution of a boundary value problem, we will be looking for
parameter values that permit the existence of solutions of special form.
Example: Buckling of a Column.
A long, slender column whose bottom end is hinged carries an axial load as
shown in Fig. 7. The upper end of the column can move up or down but not
sideways. The displacement of the column’s centerline from a vertical reference
line is given byu(x).Ifthecolumnwerecutatanypointx, an upward forceP
and a clockwise momentPu(x)wouldhavetobeappliedtotheupperpartto
keep it in equilibrium (see Fig. 8). This force and moment must be supplied
by the lower part of the column.