Chapter 6: Q35E (page 337)

Use the annihilator method to show that if $a_{0} = 0$ and $a_{1} \neq 0$ in (4) andhas the form $f (x)$ given in (17), then equation (4) has a particular solution of the form $y_{p} (x) = x {B_{m} x^{m} + B_{m - 1} x^{m - 1} + \dots + B_{1} x + B_{0}}$

Short Answer

Expert verified

$y_{p} = x (B_{m} x^{m} + \dots . + B_{0})$ is the form of particular solution.

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Definition

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

Check for particular solution

It is given that $a_{0} = 0, a_{1} \neq 0$

Equation (17) is $f (x) = b_{m} x^{m} + \dots \dots . . + b_{1} x + b_{0}$

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

For $a_{0} = 0$

$a_{n} y^{(n)} + \dots \dots + a_{1} y^{1} = f (x)$

Let $y^{'} = ω$

$(a_{n} ω^{n - 1} + \dots \dots + a_{1}) = 0$

For $y_{p} \neq y_{g}$

$y_{p} = x (B_{m} x^{m} + \dots . + B_{0})$

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Use the annihilator method to show that if $a_{0} = 0$ and $a_{1} \neq 0$ in (4) andhas the form $f (x)$ given in (17), then equation (4) has a particular solution of the form $y_{p} (x) = x {B_{m} x^{m} + B_{m - 1} x^{m - 1} + \dots + B_{1} x + B_{0}}$

Short Answer

Step by step solution

Definition

Check for particular solution

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Use the annihilator method to show that ifa0=0anda1≠0in (4) andhas the form f(x)given in (17), then equation (4) has a particular solution of the formyp(x)=x{Bmxm+Bm-1xm-1+⋯+B1x+B0}

Short Answer

Step by step solution

Definition

Check for particular solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

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Geometry

Study anywhere. Anytime. Across all devices.

Use the annihilator method to show that if $a_{0} = 0$ and $a_{1} \neq 0$ in (4) andhas the form $f (x)$ given in (17), then equation (4) has a particular solution of the form $y_{p} (x) = x {B_{m} x^{m} + B_{m - 1} x^{m - 1} + \dots + B_{1} x + B_{0}}$