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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: Q14E (page 337)

find a differential operator that annihilates the given function.
$e^{5 x}$

Short Answer

Expert verified

$D - 5$ is the differential operator that annihilates the given function.

Step by step solution

Differentiate the function continuously

Let the function be $f (x) = e^{5 x}$

Then

$f' (x) = 5 e^{5 x} = 5 f (x) 鈬� f' (x) - 5 f (x) = 0$

Hence $(D - 5) [f] = 0$

Then $D - 5$ is the differential operator that annihilates the given function.

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Most popular questions from this chapter

Constructing Differential Equations. Given three functions that $f_{1} (x), 鈥� f_{2} (x), f_{3} (x)$ are each three times differentiable and whose Wronskian is never zero on (a, b), show that the equation

$| \begin{matrix} f_{1} (x) & f_{2} (x) & f_{3} (x) & y \\ f_{1}' (x) & f_{2}' (x) & f_{3}' (x) & y' \\ f_{1}'' (x) & f_{2}'' (x) & f_{3}'' (x) & y'' \\ f_{1}''' (x) & f_{2}''' (x) & f_{3}'' (x) & y''' \end{matrix} | = 0$

is a third-order linear differential equation for which ${f_{1}, 鈥� f_{2} 鈥� f_{3}}$ is a fundamental solution set. What is the coefficient of y鈥� in this equation?

find a differential operator that annihilates the given function.

$3 x^{2} - 6 x + 1$

find a differential operator that annihilates the given function.

$x e^{3 x} c o s 5 x$

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

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

In Problems 1-6, use the method of variation of parameters to determine a particular solution to the given equation.

$y^{'''} - 3 y^{''} + 3 y^{'} - y = e^{x}$

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find a differential operator that annihilates the given function.e5x

Short Answer

Step by step solution

Differentiate the function continuously

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find a differential operator that annihilates the given function.
$e^{5 x}$