Explanation Given: The motion of a pair of identical pendulums coupled by a spring is modeled by the system are, m x 1 ′′ = − m g l x 1 − k x 1 + k x 2 m x 2 ′′ = − m g l x 2 − k x 1 + k x 2 Approach: The general solution for the system of equations is solved with the help of Elimination method. The steps are as follows: Step 1 The system is written in operator form; that is, in D , where D = d / d t . Step 2 Eliminate one of the variables, say, y , and solve the resulting equation for x ( t ) .If the system is degenerate then stop. A separate analysis is required to determine whether or not there are solutions. Step 3 If possible, use the system to derive an equation that involves y ( t ) but not its derivatives. Substitute the found expression for x ( t ) into this equation to get a formula for y ( t ) . Step 4 The expressions for x ( t ) and y ( t ) gives the desired solution. Calculation: The given system of equations is given as, ( m D 2 + ( k + m g l ) ) x 1 − k x 2 = 0 ... ( 1 ) ( m D 2 + ( k + m g l ) ) x 2 − k x 1 = 0 ... ( 2 ) Here, D = d d t . Solve the equation ( 1 ) and equation ( 2 ) as follows. Multiply equation ( 1 ) by ( m D 2 + k + m g l ) as, ( m D 2 + k + m g l ) 2 x 1 − k ( m D 2 + k + m g l ) x 2 = 0 … ( 3 ) Multiply equation ( 2 ) by k as k ( m D 2 + ( k + m g l ) ) x 2 − k 2 x 1 = 0 ( 4 ) Add equations ( 3 ) and ( 4 ) as,

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 5.6, Problem 9E

To determine

To find:

The two normal frequencies for the system.

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Chapter 5.6, Problem 8E

Chapter 5.6, Problem 10E

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Chapter 5 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

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The two normal frequencies for the system.

Solution Summary: The author explains how the system of equations is solved with the help of Elimination method.

To find: The two normal frequencies for the system.

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Chapter 5 Solutions

To find:

The two normal frequencies for the system.