note that : it asks for all periodic solutions; so i dont htink answering the general solution is enough? help me please! **Problem Statement:**Find all periodic solutions of the equation\[ y'' - 3y' + 2y = \sin x. \]**Analysis:**This is a second-order linear non-homogeneous differential equation. We seek solutions $ y(x) $ that are periodic. The equation consists of:- $ y'' $, the second derivative of $ y $,- $ y' $, the first derivative of $ y $,- $ y $ itself,- The non-homogeneous term $ \sin x $, which is sinusoidal with a known period of $ 2\pi $.To solve this, we typically find the complementary solution (solution to the associated homogeneous equation) and a particular solution (fitting the non-homogeneous part).

Given Information:The differential equation is given as .To find:All periodic solutions to the given…

Answered: Find all periodic solutions of the…

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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note that : it asks for all periodic solutions; so i dont htink answering the general solution is enough? help me please!

**Problem Statement:**

Find all periodic solutions of the equation

\[ y'' - 3y' + 2y = \sin x. \]

**Analysis:**

This is a second-order linear non-homogeneous differential equation. We seek solutions $ y(x) $ that are periodic. The equation consists of:

- $ y'' $, the second derivative of $ y $,
- $ y' $, the first derivative of $ y $,
- $ y $ itself,
- The non-homogeneous term $ \sin x $, which is sinusoidal with a known period of $ 2\pi $.

To solve this, we typically find the complementary solution (solution to the associated homogeneous equation) and a particular solution (fitting the non-homogeneous part).

Expert Solution

Step 1: Analysis and Introduction

Given Information:

The differential equation is given as $y apostrophe apostrophe minus 3 y apostrophe plus 2 y equals sin open parentheses x close parentheses$ .

To find:

All periodic solutions to the given equation.

Concept used:

The differential equation is of the form $a y apostrophe apostrophe open parentheses x close parentheses plus b y apostrophe open parentheses x close parentheses plus c y open parentheses x close parentheses equals f open parentheses x close parentheses$ has the auxiliary equation as $a m squared plus b m plus c equals 0$ .

If the auxilary equation has the real and unequal roots, then the complementary equation is $y subscript c equals c subscript 1 e to the power of m subscript 1 x end exponent plus c subscript 2 e to the power of m subscript 2 open parentheses x close parentheses end exponent$ .

The guess for the particular solution when $f open parentheses x close parentheses equals sin open parentheses x close parentheses$ is $y subscript p open parentheses x close parentheses equals A space sin open parentheses x close parentheses plus B space cos open parentheses x close parentheses$ .

The function sine and cosine are periodic in nature with period $2 pi$ .

Differentiation Formula:

$fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses equals cos open parentheses x close parentheses semicolon space fraction numerator d over denominator d x end fraction open parentheses sin open parentheses x close parentheses close parentheses equals negative cos open parentheses x close parentheses$