Explanation Given: The system of the three equation is, x ′ = 4 x − 4 z , y ′ = 4 y − 2 z , z ′ = − 2 x − 4 y + 4 z Also use elimination method to find general solution. Approach: a) Write the system of three equation. b) Express the system in operator notation. c) Eliminate z solve for x and y . d) Write general solution for x . e) Write general solution for z f) Further solve to find for the general of y . Calculation: Write the system of three equation, x ′ = 4 x − 4 z , y ′ = 4 y − 2 z , z ′ = − 2 x − 4 y + 4 z Express the system in operator notation, ( D − 4 ) [ x ] + 4 z = 0 ... ( 1 ) ( D − 4 ) [ y ] + 2 z = 0 ... ( 2 ) 2 x + 4 y + ( D − 4 ) [ z ] = 0 ... ( 3 ) From equation ( 1 ) and ( 2 ) , ( 4 − D ) x = ( 8 − 2 D ) y . ... ( 4 ) From equation ( 1 ) and ( 3 ) − ( D − 4 ) ( D − 4 ) x 4 = − 2 x − 4 y − ( D − 4 ) 2 x = − 8 x − 16 y − ( D 2 − 8 D − 8 ) x = − 16 y Thus, − ( D 2 − 8 D − 8 ) x = − 16 y ... ( 5 ) Multiply − 16 in equation 4 and ( 8 − 2 D ) in equation ( 5 ) , − 16 ( 4 − D ) x = − 16 ( 8 − 2 D ) y ( 8 − 2 D ) ( − D 2 + 8 − 8 ) x = − 16 ( 8 − 2 D ) y ( ( − 64 + 16 D ) − ( 8 − D ) ( − D 2 + 8 D − 8 ) ) x = 0 ( − 64 + 16 D + 8 D 2 − 64 D + 64 − 2 D 3 + 16 D 2 − 16 D ) x = 0 Then, ( − 2 D 3 + 24 D 2 − 64 D ) x = 0 ( D ) ( D − 4 ) ( D − 8 ) x = 0 Thus, The general solution is, x ( t ) = c 1 + c 2 e 4t + c 3 e 8t

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Videos

Question

Chapter 5.2, Problem 27E

To determine

To find:

A general solution for the system of three equations in three unknown functions x(t), y(t) and z(t).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.2, Problem 26E

Chapter 5.2, Problem 28E

Students have asked these similar questions

Please answer this correctly with complete solution please

Please answer this correctly and show me the complete solution

Who is the better student, relative to his or her classmates? Here’s all the information you ever wanted to know

Chapter 5 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

A general solution for the system of three equations in three unknown functions x ( t ) , y ( t ) and z ( t ) .

Solution Summary: The author explains how to find a general solution for the system of three equations in three unknown functions.

Videos

To find:

A general solution for the system of three equations in three unknown functions x(t), y(t) and z(t).

Want to see the full answer?

Chapter 5 Solutions

A general solution for the system of three equations in three unknown functions x ( t ) , y ( t ) and z ( t ) .

Solution Summary: The author explains how to find a general solution for the system of three equations in three unknown functions.

Videos

To find: A general solution for the system of three equations in three unknown functions x(t), y(t) and z(t).

Want to see the full answer?

Chapter 5 Solutions

To find:

A general solution for the system of three equations in three unknown functions x(t), y(t) and z(t).