Explanation Given: The given system is: X ′ ( t ) = A X ( t ) + f ( t ) Here, A = [ 1 2 3 2 ] , f ( t ) = [ 1 − 1 ] Procedure used: Variation of parameters: Suppose n × n constant matrix A has n linearly independent eigenvectors u 1 , u 2 , ⋯ , u n . Let r i be the eigenvalues corresponding to u i . Then, { e r 1 t u 1 , e r 2 t u 2 , ⋯ , e r n t u n } is a fundamental solution set and [ e r 1 t u 1 , e r 2 t u 2 , ⋯ , e r n t u n ] is called fundamental matrix. Let X ( t ) is the fundamental matrix for the non-homogeneous system X ′ ( t ) = A ( t ) X ( t ) + f ( t ) . Hence, the particular solution of non-homogeneous system is X p ( t ) = X ( t ) v ( t ) = X ( t ) ∫ X − 1 ( t ) f ( t ) d t . The general solution is the combination of X ( t ) c and X p ( t ) . x ( t ) = X ( t ) c + X p ( t ) Calculation: Consider the given matrix: A = [ 1 2 3 2 ] Substitute the value of matrix A in the characteristic equation and solve for r : | [ 1 2 3 2 ] − r [ 1 0 0 1 ] | = 0 | [ 1 2 3 2 ] − [ r 0 0 r ] | = 0 | [ 1 − r 2 3 2 − r ] | = 0 ( 1 − r ) ( 2 − r ) − 6 = 0 Further simplify as: 2 − r − 2 r + r 2 − 6 = 0 r 2 − 3 r − 4 = 0 r 2 − 4 r + r − 4 = 0 r ( r − 4 ) + 1 ( r − 4 ) = 0 Further simplify as: ( r − 4 ) ( r + 1 ) = 0 Equate first factor to 0: ( r − 4 ) = 0 r = 4 Equate second factor to 0: ( r + 1 ) = 0 r = − 1 Therefore, the eigenvalues are r 1 = − 1 and r 2 = 4 . Find the eigenvector corresponding to eigenvalue r 1 = − 1 by using the formula ( A − r I ) u = 0 : ( A − r I ) u = ( [ 1 2 3 2 ] − ( − 1 ) [ 1 0 0 1 ] ) [ u 1 u 2 ] = ( [ 1 2 3 2 ] + [ 1 0 0 1 ] ) [ u 1 u 2 ] = [ 2 2 3 3 ] [ u 1 u 2 ] The matrix equation is: [ 2 2 3 3 ] [ u 1 u 2 ] = [ 0 0 ] It can be write in scalar equation as, u 1 + u 2 = 0 . Choose the arbitrary value of u 2 = 1 , then u 1 = − 1 . The eigenvector can be expressed as: u 1 = [ − 1 1 ] Find the eigenvector corresponding to eigenvalue r 2 = 4 by using the formula ( A − r I ) u = 0 : ( A − r I ) u = ( [ 1 2 3 2 ] − 4 [ 1 0 0 1 ] ) [ u 1 u 2 ] = ( [ 1 2 3 2 ] − [ 4 0 0 4 ] ) [ u 1 u 2 ] = [ − 3 2 3 − 2 ] [ u 1 u 2 ] The matrix equation is: [ − 3 2 3 − 2 ] [ u 1 u 2 ] = [ 0 0 ] It can be write in scalar equation as, − 3 u 1 + 2 u 2 = 0

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 9.7, Problem 12E

To determine

To find:

The general solution by using variation of parameters formula for the given system.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.7, Problem 11E

Chapter 9.7, Problem 13E

Students have asked these similar questions

Using FDF, BDF, and CDF, find the first derivative; 1. The distance x of a runner from a fixed point is measured (in meters) at an interval of half a second. The data obtained is: t 0 x 0 0.5 3.65 1.0 1.5 2.0 6.80 9.90 12.15 Use CDF to approximate the runner's velocity at times t = 0.5s and t = 1.5s 2. Using FDF, BDF, and CDF, find the first derivative of f(x)=x Inx for an input of 2 assuming a step size of 1. Calculate using Analytical Solution and Absolute Relative Error: = True Value - Approximate Value| x100 True Value 3. Given the data below where f(x) sin (3x), estimate f(1.5) using Langrage Interpolation. x 1 1.3 1.6 1.9 2.2 f(x) 0.14 -0.69 -0.99 -0.55 0.31 4. The vertical distance covered by a rocket from t=8 to t=30 seconds is given by: 30 x = Loo (2000ln 140000 140000 - 2100 9.8t) dt Using the Trapezoidal Rule, n=2, find the distance covered. 5. Use Simpson's 1/3 and 3/8 Rule to approximate for sin x dx. Compare the results for n=4 and n=8

Can you check if my step is correct?

I need help explaining on this example on how can I define the Time-Domain Function, Apply the Laplace Transformation Formula, and Simplify to Find the Frequency-Domain Expression. I need to understand on finding Y(s)

Chapter 9 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.

The general solution by using variation of parameters formula for the given system.

Solution Summary: The author explains the general solution by using variation of parameters formula for the given system.

Videos

To find: The general solution by using variation of parameters formula for the given system.

Want to see the full answer?

Chapter 9 Solutions

To find:

The general solution by using variation of parameters formula for the given system.