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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: Q36E (page 338)

Use the annihilator method to show that if $f (x)$ in (4) has the form $f (x) = B e^{伪 x}$ , then equation (4) has a particular solution of the form $y_{p} (x) = x^{s} B e^{伪 x}$ , where $s$ is chosen to be the smallest nonnegative integer such that $x^{s} e^{伪 x}$ is not a solution to the corresponding homogeneous equation

Short Answer

Expert verified

$y_{p} = x^{s} 位 e^{伪 x}$ is the form of particular solution.

Step by step solution

01

Definition

A linear differential operator $A$ is said to annihilate a function $f$ if $A [f] (x) = 0 鈥� 鈥� 鈥� 鈥� 鈥� - - (2)$ for all x. That is, $A$ annihilates $f$ if $f$ is a solution to the homogeneous linear differential equation (2) on.

$(- 鈭�, 鈭�)$

02

Find particular solution

Given that $f (x) = B e^{伪 x}$

Since $e^{伪 x}$ is annihilated by $(D - 伪)$ so we have:

$(D - 伪)$ $(a_{n} y^{(n)} (x) + 鈥� . . + a_{0} y) = B e^{伪 x}$

To check if $e^{伪 x}$ is solution of homogeneous equation then $y_{p} 鈮� y_{homogeneous}$ So take .

$y_{p} = x^{s} 位 e^{伪 x}$

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Most popular questions from this chapter

. find a differential operator that annihilates the given function.

$x e^{- 2 x} + x e^{- 5 x} s i n 3 x$

Use the annihilator method to show that if $a_{0} 鈮� 0$ in equation (4) and $f (x)$ has the form (17) $f (x) = b_{m} x^{m} + b_{m - 1} x^{m - 1} + 鈰� + b_{1} x + b_{0}$ , then $y_{p} (x) = B_{m r} x^{m} + B_{m - 1} x^{m - 1} + 鈰� + B_{1} x + B_{0}$ is the form of a particular solution to equation (4).

Use the reduction of order method described in Problem 31 to find three linearly independent solutions to, given $y''' - 2 y'' + y' - 2 y = 0$ that $f (x) = e^{2 x}$ is a solution.

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

${\cos 2 x, 鈥� \cos^{2} x, 鈥� \sin^{2} x}$ on $(- 鈭�, 鈥� 鈭�)$

use the annihilator method to determinethe form of a particular solution for the given equation.

$y'' + 6 y' + 8 y = e^{3 x} - s i n x$

See all solutions

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