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Chapter 8: Problem 22
$$x y^{\prime \prime}+y^{\prime}-4 y=0$$
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\(y^{\prime \prime}-(\sin x) y=\cos x\)
$$3 x y^{\prime \prime}+2(1-x) y^{\prime}-4 y=0$$
Buckling Columns. In the study of the buckling of a column whose cross section varies, one encounters the equation $$\quad x^{n} y^{\prime \prime}(x)+\alpha^{2} y(x)=0, \quad x>0$$ where x is related to the height above the ground and y is the deflection away from the vertical. The positive constant a 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 \((45)\) with \(n=4\) to the form \(\frac{d^{2} y}{d t^{2}}+\frac{2}{t} \frac{d y}{d t}+\alpha^{2} y=0, \quad t>0\) (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 \((45) .\)
\(x(1-x) y^{\prime \prime}+(1-3 x) y^{\prime}-y=0\)
van der Pol Equation. In the study of the vacuum tube, the following equation is encountered:$$y^{\prime \prime}+(0.1)\left(y^{2}-1\right) y^{\prime}+y=0$$ Find the Taylor polynomial of degree 4 approximating the solution with the initial values \(y(0)=1\), \(y^{\prime}(0)=0\).
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