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Chapter 13: Q82SE (page 617)
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\(\int_0^x r {J_0}(r)dr = x \times {J_1}(x)\)
The obtained integral is \(\int_0^x r {J_0}(r)dr = x{J_1}(x)\).
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Buckling of a Thin Vertical Column In Example 4 of Section 5.2 we saw that when a constant vertical compressive force, or load, \(P\) was applied to a thin column of uniform cross section and hinged at both ends, the deflection \(y(x)\) is a solution of the BVP:
\(El\frac{{{d^2}y}}{{d{x^2}}} + Py = 0,y(0) = 0,y(L) = 0\)
(a)If the bending stiffness factor \(El\) is proportional to \(x\), then \(El(x) = kx\), where \(k\) is a constant of proportionality. If \(El(L) = kL = M\) is the maximum stiffness factor, then \(k = M/L\) and so \(El(x) = Mx/L\). Use the information in Problem 39 to find a solution of \(M\frac{x}{L}\frac{{{d^2}y}}{{d{x^2}}} + Py = 0,y(0) = 0,y(L) = 0\) if it is known that \(\sqrt x {Y_1}(2\sqrt {\lambda x} )\) is not zero at \(x = 0\).
(b) Use Table 6.4.1 to find the Euler load \({P_1}\) for the column.
(c) Use a CAS to graph the first buckling mode \({y_1}(x)\) corresponding to the Euler load \({P_1}\). For simplicity assume that \({c_1} = 1\) and \(L = 1\).
Use the formula obtained in Example \(6\) along with part (a) of Problem \(27\) to derive the recurrence relation \(2v{J_v}(x) = x{J_{v + 1}}(x) + x{J_{v - 1}}(x)\).
Use Table 6.4.1 to find the first three positive eigenvalues and corresponding eigenfunctions of the boundary-value problem \(xy'' + y' + \lambda xy = 0,y(x),y'(x)\) bounded as \(x \to {0^ + },y(2) = 0\). (Hint: By identifying \(\lambda = {\alpha ^2}\), the DE is the parametric Bessel equation of order zero.)
Proceed as on page \(269\) to derive the elementary form of \({J_{ - 1/2}}(x)\) given in \((27)\).
(a) Use the explicit solutions \({y_1}(x)\) and \({y_2}(x)\) of Legendre鈥檚 equation given in \((32)\) and the appropriate choice of \({c_0}\) and \({c_1}\) to find the Legendre polynomials \({P_6}(x)\) and \({P_7}(x)\).
(b) Write the differential equations for which \({P_6}(x)\) and \({P_7}(x)\) are particular solutions.
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