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### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

# Use the annihilator method to show that if${{a}}_{{0}}{\ne }{0}$in equation (4) and ${f}\left(x\right)$ has the form (17) ${f}{\left(}{x}{\right)}{=}{{b}}_{{m}}{{x}}^{{m}}{+}{{b}}_{m-1}{{x}}^{m-1}{+}{\cdots }{+}{{b}}_{{1}}{x}{+}{{b}}_{{0}}$, then ${{y}}_{{p}}{\left(}{x}{\right)}{=}{{B}}_{mr}{{x}}^{{m}}{+}{{B}}_{m-1}{{x}}^{m-1}{+}{\cdots }{+}{{B}}_{{1}}{x}{+}{{B}}_{{0}}$ is the form of a particular solution to equation (4).

${y}_{p}={B}_{m}{x}^{m}+\dots \dots \dots +{B}_{1}x+{B}_{0}$is the form of particular solution.

See the step by step solution

## Step 1: Definition

A linear differential operator ${\mathbit{A}}$is said to annihilate a function ${\mathbf{}}{\mathbit{f}}$if ${\mathbit{A}}{\mathbf{\left[}}{\mathbit{f}}{\mathbf{\right]}}{\mathbf{\left(}}{\mathbit{x}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{0}}{\mathbf{ }}{\mathbf{ }}{\mathbf{ }}{\mathbf{ }}{\mathbf{ }}{\mathbf{-}}{\mathbf{-}}{\mathbf{\left(}}{\mathbf{2}}{\mathbf{\right)}}$for all x. That is,${\mathbit{A}}$ annihilates f if fis a solution to the homogeneous linear differential equation (2) on ${\mathbf{\left(}}{\mathbf{-}}{\mathbf{\infty }}{\mathbf{,}}{\mathbf{\infty }}{\mathbf{\right)}}$.

## Step 2: Check for particular solution

It is given that $f\left(x\right)={b}_{m}{x}^{m}+\dots \dots \dots +{b}_{1}x+{b}_{0}$ and ${a}_{0}\ne 0$.

Then the ${y}_{g}$is given by:

$\left({a}_{n}{y}^{\left(n\right)}+\dots ..+{a}_{1}{y}^{\text{'}}+{a}_{0}y\right)=f$

So ${y}_{p}={B}_{m}{x}^{m}+\dots \dots \dots +{B}_{1}x+{B}_{0}$

(Then ${y}_{p}\ne {y}_{g}$)

Therefore Homogeneous auxiliary equation is not particular solution for $f$'s, annihilator.