Explanation Given: The system of the three equation is, x ′ = x + 2 y + z , y ′ = 6 x − y , z ′ = − x − 2 y − 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 y . e) Write general solution for x f) Further solve to find for the general of z . Calculation: Write the system of three equation, x ′ = x + 2 y + z , y ′ = 6 x − y z ′ = − x − 2 y − z , Express the system in operator notation, ( D − 1 ) [ x ] − 2 y − z = 0 ... ( 1 ) 6 x − ( 1 + D ) [ y ] = 0 ... ( 2 ) x + 2 y + ( D + 1 ) [ z ] = 0 ... ( 3 ) Multiply ( D + 1 ) in equation ( 1 ) and add with equation ( 3 ) , ( D 2 − 1 ) x − ( 2 D + 2 ) y − ( D + 1 ) z + x + 2 y + ( D + 1 ) z = 0 Write, D 2 x − 2 D y = 0 . ... ( 4 ) Put equation ( 2 ) , 6 x − ( 1 + D ) y = 0 . Multiply 6 in equation ( 4 ) and D 2 in equation ( 2 ) , 6 D 2 x − 12 D y = 0 6 D 2 x − D 2 ( 1 + D ) y = 0 Subtract above equations, Put, D ( D + D 2 − 12 ) y = 0 The auxiliary equation is, m ( m + m 2 − 12 ) = 0 m ( m + 4 ) ( m − 3 ) = 0 m = − 4 , 0 , 3 Thus the general solution for y ( t ) is, y ( t ) = c 1 + c 2 e 3t + c 3 e − 4t . Differentiate y ( t ) with respect to t , y ′ ( t ) = + 3 c 2 e 3t − 4 c 3 e − 4t Take, y ′ = 6 x − y from equation ( 2 ) and solve or x . Put values,

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ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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Chapter 5.2, Problem 28E

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

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Chapter 5.2, Problem 27E

Chapter 5.2, Problem 29E

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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.

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To find:

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

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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.

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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).