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Found in: Page 338

### 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${\mathbit{f}}{\mathbf{\left(}}{\mathbit{x}}{\mathbf{\right)}}$in (4) has the form ${\mathbit{f}}{\mathbf{\left(}}{\mathbit{x}}{\mathbf{\right)}}{\mathbf{=}}{\mathbit{a}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}{\mathbit{\beta }}{\mathbit{x}}{\mathbf{+}}{\mathbit{b}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}{\mathbit{\beta }}{\mathbit{x}}{\mathbf{,}}$then equation (4) has a particular solution of the form(18)${{\mathbit{y}}}_{{\mathbf{p}}}{\mathbf{\left(}}{\mathbit{x}}{\mathbf{\right)}}{\mathbf{=}}{{\mathbit{x}}}^{{\mathbf{s}}}{\mathbf{\left\{}}{\mathbit{A}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}{\mathbit{\beta }}{\mathbit{x}}{\mathbf{+}}{\mathbit{B}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}{\mathbit{\beta }}{\mathbit{x}}{\mathbf{\right\}}}$ ,where s is chosen to be the smallest nonnegative integer such that ${{\mathbit{x}}}^{{\mathbf{3}}}{\mathbit{c}}{\mathbit{o}}{\mathbit{s}}{\mathbit{\beta }}{\mathbit{x}}$ and ${{\mathbit{x}}}^{{\mathbf{3}}}{\mathbf{sin}}{\mathbit{\beta }}{\mathbit{x}}$are not solutions to the corresponding homogeneous equation

${y}_{p}={x}^{s}\left(A\mathrm{cos}\beta x+B\mathrm{sin}\beta x\right)$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${\mathbit{f}}$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. That is, ${\mathbit{A}}$ annihilates ${\mathbit{f}}$if ${\mathbit{f}}$is a solution to the homogeneous linear differential equation (2) on ${\mathbf{\left(}}{\mathbf{-}}{\mathbf{\infty }}{\mathbf{,}}{\mathbf{\infty }}{\mathbf{\right)}}$.

## Step 2: For particular solution

Equation (4) is given by ${a}_{n}{y}^{\left(n\right)}+{a}_{n-1}{y}^{n-1}+\dots \dots ..+{a}_{0}y=f$

Also given $f\left(x\right)=a\mathrm{cos}\beta x+b\mathrm{sin}\beta x$

Then, $\left({a}_{n}{D}^{n}+{a}_{n-1}{D}^{n-1}+\dots \dots ..+{a}_{0}\right)y=a\mathrm{cos}\beta x+b\mathrm{sin}\beta x$

So, $\mathrm{sin}\beta x\mathrm{cos}\beta x$ is annihilated by$\left({D}^{2}+{\beta }^{2}\right)\text{}$

So, $\left({D}^{2}+{\beta }^{2}\right)\left({a}_{n}{D}^{n}+{a}_{n-1}{D}^{n-1}+\dots \dots ..+{a}_{0}\right)y=0$

For particular solution i.e; ${y}_{p}$ check if $\mathrm{cos}\beta x,\mathrm{sin}\beta x$are solutions to homogeneous, particular solution is different than homogeneous solution choose

${y}_{p}={x}^{s}\left(A\mathrm{cos}\beta x+B\mathrm{sin}\beta x\right)$