Question

Use the annihilator method to show that if${\mathit{a}}_{\mathbf{0}}\mathbf{=}\mathbf{0}$and${\mathit{a}}_{\mathbf{1}}\mathbf{\ne }\mathbf{0}$in (4) andhas the form $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$given in (17), then equation (4) has a particular solution of the form${\mathit{y}}_{\mathbf{p}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{x}\left\{B_m{x}^m+B_{m-1}{x}^{m-1}+\cdots +B_{1}x+B_0\right\}$

Short answer

$y_p=x\left(B_m{x}^m+\dots .+B_0\right)$is the form of particular solution.

Step-by-step solution

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 x. That is, $\mathit{A}$ annihilates $\mathit{f}$if$\mathit{f}$is a solution to the homogeneous linear differential equation (2) on$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$.

Check for particular solution

It is given that$a_0=0,a_1\ne 0$

Equation (17) is$f\left(x\right)=b_m{x}^m+\dots \dots ..+b_{1}x+b_0$

Then,$a_n{y}^{\left(n\right)}+\dots \dots +a_1{y}^1+a_{0}y=f\left(x\right)$

For$a_0=0$

$a_n{y}^{\left(n\right)}+\dots \dots +a_1{y}^1=f\left(x\right)$

Let$y^{\text{'}}=\omega$

$\left(a_n{\omega }^{n-1}+\dots \dots +a_1\right)=0$

For$y_p\ne y_g$

$y_p=x\left(B_m{x}^m+\dots .+B_0\right)$