Please answer ASAP for like Thanks Problem 5: Consider the following nonlinear differential equation:d²xdx+ 3x + 2xdtf(x).dt²If f(x) = sin(x), linearize the differential equation around x = π.=

The given differential equation is We have to linearize the given diffetrential equation at…

Answered: Problem 5: Consider the following…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 5: Consider the following nonlinear differential equation:
d²x
dx
+ 3x + 2x
dt
f(x).
dt²
If f(x) = sin(x), linearize the differential equation around x = π.
=

Expert Solution

Step 1: ''Introduction to the solution''

The given differential equation is $left parenthesis x apostrophe apostrophe plus 3 x x apostrophe plus 2 x right parenthesis equals sin x......... left parenthesis 1 right parenthesis space$

We have to linearize the given diffetrential equation at $x equals pi.$

Suppose that $x almost equal to x plus increment x comma$ $increment x rightwards arrow$ small deviation from $x.$

Then, $x apostrophe equals increment x apostrophe space space a n d space x apostrophe apostrophe equals increment x apostrophe apostrophe$

Putting these values in (1) , we obtain, $increment x apostrophe apostrophe plus 3 left parenthesis straight pi plus increment x right parenthesis increment x apostrophe plus 2 open parentheses straight pi plus increment straight x close parentheses equals sin open parentheses pi plus increment x close parentheses$

$rightwards double arrow increment x apostrophe apostrophe plus 3 straight pi increment straight x apostrophe plus 2 increment straight x equals negative 2 straight pi plus sin straight pi plus increment xcos straight pi$ [ignoring higher order terms]

$rightwards double arrow increment x apostrophe apostrophe plus 3 straight pi increment straight x apostrophe plus 2 increment straight x equals negative 2 straight pi minus increment straight x$

$rightwards double arrow increment x apostrophe apostrophe plus 3 straight pi increment straight x apostrophe plus 3 increment straight x equals negative 2 straight pi.......... left parenthesis 2 right parenthesis$