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Chapter 8: Problem 37

$$6 x^{3} y^{\prime \prime \prime}+13 x^{2} y^{\prime \prime}-\left(x^{2}+3 x\right) y^{\prime}-x y=0$$

Short Answer

Expert verified
The given third order non-homogeneous differential equation has no general solving method. Solutions would require either reducing the order (which in this case is not straightforward), or numerical methods. Unfortunately, this task is beyond the scope of most high school curricula.

Step by step solution

01

Identify the differential equation

The given exercise is a third order differential equation form, \( a(x) y''' + b(x) y'' - c(x) y' - f(x) y = 0 \), where \( a(x) = 6x^3 \), \( b(x) = 13x^2 \), \( c(x) = x^2 + 3x \), and \( f(x) = x \).
02

Simplify the differential equation

The equation cannot be simplified and a general form of third order differential equation has no general solution method. It can either be solved by using methods like guessing the solution, reducing the order or by using numerical methods.
03

Try a method for the solution

In this case, guessing the solution (a method often taught in introductions to differential equation courses) is unlikely to result in any progress. The other two mentioned options would be to reduce the order of the differential equation or the use of numerical methods. For the first option, it would be needed to find a solution of the associated second order differential equation and then use variation of parameters to find a solution of the original equation. These methods are often difficult and can lead to a solution more by luck than by a systematic approach.

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Most popular questions from this chapter

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 \((45) .\)

\(x^{2} y^{\prime \prime}+x y^{\prime}+x^{2} y=0\)

Let \(f(x)=\left\\{\begin{array}{ll}{e^{-1 / x^{2}},} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.\) Show that \(f^{(n)}(0)=0\) for \(n=0,1,2, \ldots\) and hence that the Maclaurin series for \(f(x)\) is \(0+0+0+\cdots\) which converges for all \(x\) but is equal to \(f(x)\) only when \(x=0 .\) This is an example of a function possessing derivatives of all orders \(\left(\) at \(x_{0}=0\right),\) whose Taylor series converges, but the Taylor series (about \(x_{0}=0 )\) does not converge to the original function! Consequently, this function is not analytic at \(x=0 .\)

$$6 x^{3} y^{\prime \prime \prime}+\left(13 x^{2}-x^{3}\right) y^{\prime \prime}+x y^{\prime}-x y=0$$

$$x(x+1) y^{\prime \prime}+(x+5) y^{\prime}-4 y=0$$
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