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

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

7th Edition

ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

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

Is it possible to show me how to come up with an exponential equation by showing all the steps work and including at least one mistake that me as a person can make. Like a calculation mistake and high light what the mistake is. Thanks so much.

iid 1. The CLT provides an approximate sampling distribution for the arithmetic average Ỹ of a random sample Y₁, . . ., Yn f(y). The parameters of the approximate sampling distribution depend on the mean and variance of the underlying random variables (i.e., the population mean and variance). The approximation can be written to emphasize this, using the expec- tation and variance of one of the random variables in the sample instead of the parameters μ, 02: YNEY, · (1 (EY,, varyi n For the following population distributions f, write the approximate distribution of the sample mean. (a) Exponential with rate ẞ: f(y) = ß exp{−ßy} 1 (b) Chi-square with degrees of freedom: f(y) = ( 4 ) 2 y = exp { — ½/ } г( (c) Poisson with rate λ: P(Y = y) = exp(-\} > y! y²

2. Let Y₁,……., Y be a random sample with common mean μ and common variance σ². Use the CLT to write an expression approximating the CDF P(Ỹ ≤ x) in terms of µ, σ² and n, and the standard normal CDF Fz(·).

Chapter 5 Solutions

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

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).