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Buckling of a Tapered Column In Example 4 of Section 3.9, we saw that when a constant vertical compressive force or load P was applied to a thin column of uniform cross section, the deflection y(x) satisfied the boundary-value problem
The assumption here is that the column is hinged at both ends. The column will buckle or deflect only when the compressive force is a critical load pn.
(a) In this problem let us assume that the column is of length L, is hinged at both ends, has circular cross sections, and is tapered as shown in FIGURE 5.2.1(a). If the column, a truncated cone, has a linear taper y = ex as shown in cross section in Figure 5.2.1(b), the moment of inertia of a cross section with respect to an axis perpendicular to the xy-plane is 
 we can write 
 Substituting J(x) into the differential equation in (24), we see that the deflection in this case is determined from the BVP
Where 
 Use the results of Problem 33 to find the critical loads Pn for the tapered column. Use an appropriate identity to express the buckling modes yn(x) as a single function.
(b) Use a CAS to plot the graph of the first buckling mode y1(x) corresponding to the Euler load P1Â when b = 11 and a= 1.
Example 4
Solve xy" + y = 0.
Problem 33
(a) The differential equation x4y'' + λy = 0 has an irregular singular point at x = 0. Show that the substitution t = 1/x yields the differential equation
which now has a regular singular point at t = 0.
(b) Use the method of this section to find two series solutions of the second equation in part (a) about the singular point t = 0.
(c) Express each series solution of the original equation in terms of elementary functions.
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