Chapter 8.7, Problem 22E

In Problem $21$ consider a column with a rectangular cross section with two sides constant and the other two changing linearly [see Figure $8.13$(b)]. In this case, $n=1$. Find at least the first four nonzero terms in the series expansion about $x=0$ for a general solution to equation $\left(45\right)$ when $n=1$.

$21$.Buckling Columns. In the study of the buckling of a column whose cross section varies, one encounters the equation

$x^n{y}^{″}\left(x\right)+{\alpha }^{2}y\left(x\right)=0$, $x+0$, $\left(45\right)$

where $\text{x}$ is related to the height above the ground and $y$ is the deflection away from the vertical. The positive constant $\alpha$ depends on the rigidity of the column, its moment of inertia at the top, and the load. The positive integer $n$ depends on the type of column. For example, when the column is a truncated cone [see Figure $8.13$(a) on page $474$], we have $n=4$.

a. Use the substitution $x=t^{-1}$ to reduce $\left(45\right)$ with $n=4$ to the form

$\frac{{\partial }^{2}y}{\partial t^2}+\frac{2}{t}\frac{dy}{dt}+{\alpha }^{2}y=0$, $t>0$.

b. Find at least the first six nonzero terms in the series expansion about $t=0$ for a general solution to the equation obtained in part (a).

c. Use the result of part (b) to give an expansion about $x=\infty$ for a general solution to $\left(45\right)$.