Chapter 6: Q36E (page 338)

Use the annihilator method to show that if $f (x)$ in (4) has the form $f (x) = B e^{α x}$ , then equation (4) has a particular solution of the form $y_{p} (x) = x^{s} B e^{α x}$ , where $s$ is chosen to be the smallest nonnegative integer such that $x^{s} e^{α x}$ is not a solution to the corresponding homogeneous equation

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$y_{p} = x^{s} λ e^{α x}$ 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.

$(- \infty, \infty)$

Find particular solution

Given that $f (x) = B e^{α x}$

Since $e^{α x}$ is annihilated by $(D - α)$ so we have:

$(D - α)$ $(a_{n} y^{(n)} (x) + \dots . . + a_{0} y) = B e^{α x}$

To check if $e^{α x}$ is solution of homogeneous equation then $y_{p} \neq y_{homogeneous}$ So take .

$y_{p} = x^{s} λ e^{α x}$

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Use the annihilator method to show that if f(x)in (4) has the formf(x)=Beαx, then equation (4) has a particular solution of the formyp(x)=xsBeαx, wheresis chosen to be the smallest nonnegative integer such thatxseαxis not a solution to the corresponding homogeneous equation

Short Answer

Step by step solution

Definition

Find particular solution

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Probability and Statistics

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Applied Mathematics

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