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Fundamentals Of Differential Equations And Boundary Value Problems

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

$Math Studyset Vaia Explanations$ Math

9th Edition · 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

01

Differentiate the function continuously

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

Then

$f' (x) = 5 e^{5 x} = 5 f (x) \Rightarrow f' (x) - 5 f (x) = 0$

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

Find a general solution for the given

linear system using the elimination method of Section 5.2.

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

Solve the given initial value problem

$\begin{array}{l} y''' + 7 y'' + 14 y' + 8 y = 0 \\ y (0) = 1 \\ y' (0) = - 3 \\ y'' (0) = 13 \end{array}$

Use the annihilator method to determine the form of a particular solution for the given equation.

(a) $y^{''} + 6 y^{'} + 5 y = e^{- x} + x^{2} - 1$

(b) $y^{'''} + 2 y^{''} - 19 y^{'} - 20 y = x e^{- x}$

(c) $y^{(4)} + 6 y^{''} + 9 y = x^{2} - s i n 3 x$

(d) $y^{'''} - y^{''} + 2 y = x s i n x$

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.

$y''' - \sqrt{x} y = sinx y (π) = 0, y' (π) = 11, y'' (π) = 3$

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

$y^{'''} - 2 y^{''} + y^{'} = x$

See all solutions

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