Explanation Given: The given system is X ′ = ( 1 0 0 2 ) X + ( 3 − 1 ) . Definition used: For any matrix A of order n × n , the expansion of e A t in power of the matrix A is e A t = 1 + A t + A 2 2 ! t 2 + A 3 3 ! t 3 + … + A k k ! t k + … . The general solution of the system X ′ = A X + F ( t ) is given by X ( t ) = e A t C + e A t ∫ t 0 t e − A s F ( s ) d s where C is constant. Calculation: In the given system, the matrix is A = ( 1 0 0 2 ) and F ( t ) = ( 3 − 1 ) . Compute A 2 by multiplying matrix A with itself. A 2 = A × A = ( 1 0 0 2 ) ⋅ ( 1 0 0 2 ) = ( 1 0 0 4 ) Find the matrix A 3 by multiplying matrix A 2 with matrix A . A 3 = A 2 ⋅ A = ( 1 0 0 4 ) ⋅ ( 1 0 0 2 ) = ( 1 0 0 8 ) Multiply the matrix A 3 with the matrix A to obtain the value of the matrix A 4 . A 4 = A 3 ⋅ A = ( 1 0 0 8 ) ⋅ ( 1 0 0 2 ) = ( 1 0 0 16 ) From the above calculation, the matrices can be written as follows. A = ( 1 0 0 2 1 ) , A 2 = ( 1 0 0 2 2 ) , A 3 = ( 1 0 0 2 3 ) , A 4 = ( 1 0 0 2 4 ) , … Proceed in this way, the general value of a matrix raised to the power of a natural number is obtained as, A k = ( 1 0 0 2 k ) for k = 1 , 2 , 3 , … Thus, the value of the expansion e A t in power of the matrix A is e A t = 1 + A 1 ! t + A 2 2 ! t 2 + A 3 3 ! t 3 + … Substitute the value of obtained matrices in above expression, e A t = ( 1 0 0 1 ) + 1 1 ! ( 1 0 0 2 ) t + 1 2 ! ( 1 0 0 4 ) t 2 + 1 3 ! ( 1 0 0 8 ) t 3 + … = ( 1 + t + t 2 2 ! + t 3 3 ! + … 0 0 1 + 2 t + ( 2 t ) 2 2 ! + ( 2 t ) 3 3 ! + … ) = ( e t 0 0 e 2 t ) Substitute – t for t in the expansion of e A t and obtain e − A t . e − A t = 1 + A 1 ! ( − t ) + A 2 2 ! ( − t ) 2 + A 3 3 ! ( − t ) 3 + … = ( 1 0 0 1 ) − 1 1 ! ( 1 0 0 2 ) t + 1 2 ! ( 1 0 0 4 ) t 2 − 1 3 ! ( 1 0 0 8 ) t 3 + … = ( 1 − t + t 2 2 ! − t 3 3 ! + … 0 0 1 − 2 t + ( 2 t ) 2 2 ! − ( 2 t ) 3 3 ! + … ) = ( e − t 0 0 e − 2 t ) The general solution is given by, X ( t ) = e A t C + e A t ∫ t 0 t

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 8.4, Problem 14E

To determine

To find: The solution of the given system subject to the initial condition X(0).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.4, Problem 13E

Chapter 8.4, Problem 15E

Students have asked these similar questions

Find the general solution of the given system. 6 -(-1)x x' = -6 11 x(t) = x(t) = e5t)*[(c1 + c2(t− 1/6))(c1 + c2t)] Your answer canno

You have been hired as an intern to run analyses on the data and report the results back to Sarah; the five questions that Sarah needs you to address are given below. please do it step by step Does there appear to be a positive or negative relationship between price and screen size? Use a scatter plot to examine the relationship. Determine and interpret the correlation coefficient between the two variables. In your interpretation, discuss the direction of the relationship (positive, negative, or zero relationship). Also discuss the strength of the relationship. Estimate the relationship between screen size and price using a simple linear regression model and interpret the estimated coefficients. (In your interpretation, tell the dollar amount by which price will change for each unit of increase in screen size). Include the manufacturer dummy variable (Samsung=1, 0 otherwise) and estimate the relationship between screen size, price and manufacturer dummy as a multiple linear…

Exercises: Find all the whole number solutions of the congruence equation. 1. 3x 8 mod 11 2. 2x+3= 8 mod 12 3. 3x+12= 7 mod 10 4. 4x+6= 5 mod 8 5. 5x+3= 8 mod 12

Chapter 8 Solutions

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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.

The solution of the given system subject to the initial condition X ( 0 ) .

Solution Summary: The author explains the solution of the given system subject to the initial condition X(0).

Concept explainers

Videos

To find: The solution of the given system subject to the initial condition X(0).

Want to see the full answer?

Chapter 8 Solutions