(a) To determine The fundamental matrix for the given system of equations. Explanation Consider the given system of equations as x ′ = ( 3 − 2 2 − 2 ) x . Note that, a single vector equation is x ′ = A x , where A is coefficient matrix . The coefficient matrix of the corresponding systems of differential equations is A = ( 3 − 2 2 − 2 ) . Characteristic equation of the coefficient matrix A = ( 3 − 2 2 − 2 ) is det ( A − λ I ) = 0 , where λ eigenvalue and I is 2 × 2 identity matrix . Simplify for the det ( A − λ I ) = 0 using the coefficient matrix as follows. det ( A − λ I ) = 0 det ( ( 3 − 2 2 − 2 ) − λ ( 1 0 0 1 ) ) = 0 det ( ( 3 − λ − 2 2 − 2 − λ ) ) = 0 ( 3 − λ ) ( − 2 − λ ) + 4 = 0 Further simplified as follows. λ 2 − λ − 2 = 0 λ 2 − 2 λ + λ − 2 = 0 ( λ − 2 ) ( λ + 1 ) = 0 λ − 2 = 0 or λ + 1 = 0 λ = 2 or λ = − 1 Compute the eigenvector x ( 1 ) = [ t 1 t 2 ] corresponding to the eigenvalue λ = 2 as follows. A x − ( 2 ) x = 0 [ 1 − 2 2 − 4 ] [ t 1 t 2 ] = [ 0 0 ] [ t 1 − 2 t 2 2 t 1 − 4 t 2 ] = [ 0 0 ] t 1 − 2 t 2 = 0 and 2 t 1 − 4 t 2 = 0 t 1 2 = t 2 1 Hence, the possible vector corresponding to the eigenvalue λ = 2 is x ( 1 ) = [ 2 1 ] (b) To determine The fundamental matrix Φ ( t ) that satisfies the given initial condition.

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Question

Chapter 7.7, Problem 1P

(a)

To determine

The fundamental matrix for the given system of equations.

(b)

To determine

The fundamental matrix Φ(t) that satisfies the given initial condition.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.6, Problem 31P

Chapter 7.7, Problem 2P

Students have asked these similar questions

Q2 29 2 Points Q2.1 1 Point [120] Let A 120 and v = 001 v is an element of Col A. v is not an element of Col A. Save Answer Q2.2 1 Point [12 0 Let A 120 and w= [001] w is an element of Nul A. Ow is not an element of Nul A. Save Answer 1 Select the statement that is true. 0 Select the statement that is true.

Q3 3 Points In the vector space V = P₂ of all polynomials of degree at most 2, consider the elements Pi(t)=1+t, P2(t) = 1-t, Ps(t)=1+2t+t². Determine if it is possible to write the polynomial q(t) = t² as a linear combination of Pi(t), p2(t), ps(t). If possible, show which linear combination of p₁(t), p2(t), ps(t) is equal to q(t) . If not possible, explain why. Show all of your work. Please select file(s) Select file(s) Save Answer

2e dx e2x ex+1

Chapter 7 Solutions

Elementary Differential Equations

Ch. 7.1 - In each of Problems 1 through 4, transform the...Ch. 7.1 - Prob. 2P Ch. 7.1 - Prob. 3P Ch. 7.1 - In each of Problems 1 through 4, transform the...Ch. 7.1 - Prob. 5P Ch. 7.1 - In each of Problems 5 and 6, transform the given...Ch. 7.1 - Systems of first order equations can sometimes be...Ch. 7.1 - Prob. 8P Ch. 7.1 - Prob. 9P Ch. 7.1 - In each of Problems 8 through 12, proceed as in...

Knowledge Booster

$Advanced Math$

Learn more about

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