Q37E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Use the annihilator method to show that if$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$in (4) has the form $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{a}\mathit{c}\mathit{o}\mathit{s}\mathit{\beta }\mathit{x}\mathbf{+}\mathit{b}\mathit{s}\mathit{i}\mathit{n}\mathit{\beta }\mathit{x}\mathbf{,}$

then equation (4) has a particular solution of the form

(18)${\mathit{y}}_{\mathbf{p}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{s}}\mathbf{\{}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{\beta }\mathit{x}\mathbf{+}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\beta }\mathit{x}\mathbf{\}}$ ,where s is chosen to be the smallest nonnegative integer such that ${\mathit{x}}^{\mathbf{3}}\mathit{c}\mathit{o}\mathit{s}\mathit{\beta }\mathit{x}$ and ${\mathit{x}}^{\mathbf{3}}\mathbf{sin}\mathit{\beta }\mathit{x}$are not solutions to the corresponding homogeneous equation

Step-by-Step Solution

Answer

$y_p=x^s(A\cos \beta x+B\sin \beta x)$is the form of particular solution.

Step 1: Definition

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

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 $f\left(x\right)=a\cos \beta x+b\sin \beta x$

Then, $\left(a_n{D}^n+a_{n-1}{D}^{n-1}+\dots \dots ..+a_0\right)y=a\cos \beta x+b\sin \beta x$

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

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 $\cos \beta x,\sin \beta x$are solutions to homogeneous, particular solution is different than homogeneous solution choose

$y_p=x^s(A\cos \beta x+B\sin \beta x)$