Explanation Approach: To find the general solution for the given differential equation, do the following. Solve the auxiliary equation for the given differential equation 1. Factorize the auxiliary equation Then solve the auxiliary equation by factorization method General equation for the complex roots is as given below. y = c 1 e r 1 x + c 2 e r 2 x + c 3 e r 3 x + c 4 e r 4 x Here, r 1 , r 2 , r 3 , and r 4 are the roots of the auxiliary equation Use the Euler’s property and find the roots of auxiliary equation. e α + i β = e α cos β + i e α sin β Calculation: Consider the differential equation: y ( 4 ) + 2 y ( 3 ) + 4 y ″ + 3 y ′ + 2 y = 0 d 4 y d x 4 + 2 d 3 y d x 3 + 4 d 2 y d x 2 + 3 d y d x + 2 y = 0 ( d 4 d x 4 + 2 d 3 d x 3 + 4 d 2 d x 2 + 3 d d x + 2 ) y = 0 Let r = d d x The auxiliary equation of the above equation is, r 4 + 2 r 3 + 4 r 2 + 3 r + 2 = 0 By factorization, r 4 + 2 r 3 + 4 r 2 + 3 r + 2 = 0 r 4 + r 3 + r 2 + r 3 + r 2 + r + 2 r 2 + 2 r + 2 = 0 r 2 ( r 2 + r + 1 ) + r ( r 2 + r + 1 ) + 2 ( r 2 + r + 1 ) = 0 ( r 2 + r + 1 ) ( r 2 + r + 2 ) = 0 The roots for r 2 + r + 1 = 0 are, r = − 1 ± 1 − 4 2 = − 1 ± 3 i 2 The roots for r 2 + r + 2 = 0 are, r = − 1 ± 1 − 8 2 = − 1 ± 7 i 2 Therefore, the roots of auxiliary equation are as follows

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 6.2, Problem 29E

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The general solution of the differential equation y(4)+2y(3)+4y″+3y′+2y=0.

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Chapter 6.2, Problem 28E

Chapter 6.2, Problem 30E

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The general solution of the differential equation y ( 4 ) + 2 y ( 3 ) + 4 y ″ + 3 y ′ + 2 y = 0 .

Solution Summary: The author explains how to find the general solution of the given differential equation by factorizing the auxiliary equation.

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To find:

The general solution of the differential equation y(4)+2y(3)+4y″+3y′+2y=0.

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The general solution of the differential equation y ( 4 ) + 2 y ( 3 ) + 4 y ″ + 3 y ′ + 2 y = 0 .

Solution Summary: The author explains how to find the general solution of the given differential equation by factorizing the auxiliary equation.

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To find: The general solution of the differential equation y(4)+2y(3)+4y″+3y′+2y=0.

Want to see the full answer?

Chapter 6 Solutions

To find:

The general solution of the differential equation y(4)+2y(3)+4y″+3y′+2y=0.