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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 6: Q20E (page 337)

find a differential operator that annihilates the given function
$x^{2} e^{x} - x s i n 4 x + x^{3}$

Short Answer

Expert verified

$D^{4} (D - 1)^{3} {(D^{2} + 16)}^{2}$ is the differential operator that annihilates the given function.

Step by step solution

01

Any nonhomogeneous term of the form f(x)=xkeαxcosβx OR role="math" localid="1663946799201" xkeαxsinβxsatisfiesrole="math" localid="1663946761280" (D-α)2+β2m[f]=0for K=0,1,2,...,m-1

Let the function be $f (x) = x^{2} e^{x} - x \sin 4 x + x^{3}$

Let $g (x) = x^{2} e^{x}$

Then

$(D - 1)^{3} [g] = 0$

Let $h (x) = x e^{- 5 x} \sin 3 x$

Then

${(D^{2} + 4^{2})}^{2} [h] = 0$

Let $i (x) = x^{3}$

Then

$D^{4} [i] = 0$

Hence

$(D - 1)^{3} {(D^{2} + 16)}^{2} D^{4} [f] = 0 鈬� D^{4} (D - 1)^{3} {(D^{2} + 16)}^{2} [f] = 0$

Then $D^{4} (D - 1)^{3} {(D^{2} + 16)}^{2}$ is the differential operator that annihilates the given function.

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Most popular questions from this chapter

Find a general solution for the given

linear system using the elimination method of Section 5.2.

$\begin{array}{l} \frac{d^{3} x}{d t^{3}} 鈭� x + \frac{d y}{d t} + y = 0 \\ \frac{d x}{d t} 鈭� x + y = 0 \end{array}$

Find a general solution to

$y^{(4)} + 2 y''' 鈭� 3 y'' 鈭� y' + \frac{1}{2} y = 0$

by using Newton鈥檚 method (Appendix B) or some othernumerical procedure to approximate the roots of the auxiliaryequation.

Higher-Order Cauchy鈥揈uler Equations. A differential equation that can be expressed in the form

$a_{n} x^{n} y^{(n)} (x) + a_{n 鈭� 1} x^{n 鈭� 1} y^{(n 鈭� 1)} (x) + . .. + a_{0} y (x) = 0$

where $a_{n}, a_{n 鈭� 1}, . ..., a_{0}$ are constants, is called a homogeneous Cauchy鈥揈uler equation. (The second-order case is discussed in Section 4.7.) Use the substitution $y = x^{y}$ to help determine a fundamental solution set for the following Cauchy鈥揈uler equations:

(a) $x^{3} y''' + x^{2} y'' 鈭� 2 x y' + 2 y = 0, x > 0$

(b) $x^{4} y^{(4)} + 6 x^{3} y''' + 2 x^{2} y'' 鈭� 4 x y + 4 y = 0, x > 0$

(c) $x^{3} y''' 鈭� 2 x^{2} y'' + 13 x y' 鈭� 13 y = 0, x > 0$

[Hint: $x^{伪 + 尾 i} = e^{(伪 + 尾 i) l n x} = x^{a} {c o s (尾 l n x) + i s i n (尾 l n x)}$ ]

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.

${x^{2}, 鈥� x^{2} - 1, 鈥� 5}$ on $(- 鈭�, 鈥� 鈭�)$

Given that ${e^{x}, e^{- x}, e^{2 x}}$ is a fundamental solution set for the homogeneous equation corresponding to the equation $y^{'''} - 2 y^{''} - y^{'} + 2 y = g (x),$

determine a formula involving integrals for a particular solution.

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