Explanation Given: The given system of differential equations, x ′ 1 = − 12 x 1 − 7 x 2 and x ′ 2 = 16 x 1 + 10 x 2 . Approach: Let, the differential equations are, x ′ 1 = a x 1 + b x 2 x ′ 2 = c x 1 + d x 2 . The given system can be written as, x ′ = A x Here, matrix A is, A = [ a b c d ] The transformed system is, y ′ = ( S − 1 A S ) y Where, x = S y To evaluate the matrix S , it is necessary to find out the eigenvalues and eigenvectors of A . Calculation: Consider the given system of differential equations, x ′ 1 = − 12 x 1 − 7 x 2 x ′ 2 = 16 x 1 + 10 x 2 . The given system can be written as, x ′ = A x [ x ′ 1 x ′ 2 ] = [ − 12 − 7 16 10 ] [ x 1 x 2 ] Here, matrix A is, A = [ − 12 − 7 16 10 ] Consider the matrix A , A = [ − 12 − 7 16 10 ] The characteristic equation of A is given as det ( A − λ I ) = 0 . det ( A − λ I ) = 0 | [ − 12 − 7 16 10 ] − [ λ 0 0 λ ] | = 0 | [ − 12 − λ − 7 16 10 − λ ] | = 0 Solve the above equation for the eigenvalues ( λ ) of A . | − 12 − λ − 7 16 10 − λ | = 0 ( − 12 − λ ) ( 10 − λ ) − ( 16 ) ( − 7 ) = 0 − 120 + 12 λ − 10 λ + λ 2 + 112 = 0 λ 2 + 2 λ − 8 = 0 Further solve the above equation. λ 2 + 4 λ − 2 λ − 8 = 0 λ ( λ + 4 ) − 2 ( λ + 4 ) = 0 ( λ − 2 ) ( λ + 4 ) = 0 λ = 2 , − 4 Therefore, the eigenvalue of A are 2 and − 4 . The eigenvector corresponding to the eigenvalue λ 1 = 2 is determined based on the linear system given as, ( A − λ I ) v = 0 ( [ − 12 − 7 16 10 ] − [ λ 0 0 λ ] ) [ v 1 v 2 ] = [ 0 0 ] [ − 12 − λ − 7 16 10 − λ ] [ v 1 v 2 ] = [ 0 0 ] Substitute 2 for λ in the above equation. [ − 12 − ( 2 ) − 7 16 10 − ( 2 ) ] [ v 1 v 2 ] = [ 0 0 ] [ − 12 − 2 − 7 16 10 − 2 ] [ v 1 v 2 ] = [ 0 0 ] [ − 14 − 7 16 8 ] [ v 1 v 2 ] = [ 0 0 ] The augmented matrix of the above expression is, [ − 14 − 7 16 8 | 0 0 ] Apply R 1 → − 1 7 R 1 .and R 2 → 1 8 R 2 . [ 2 1 2 1 | 0 0 ] Apply R 2 → R 2 − R 1 . [ 2 1 0 0 | 0 0 ] Thus, [ 2 1 0 0 ] [ v 1 v 2 ] = [ 0 0 ] [ 2 v 1 + v 2 0 ] = [ 0 0 ] The equation obtained is, 2 v 1 + v 2 = 0 Solve the above equation for v 2 . v 2 = − 2 v 1 Let, v 1 = k , then v 2 = − 2 k . Thus, v 1 = k and v 2 = − 2 k , the general solution to this system is, v = [ v 1 , v 2 ] T = [ k , − 2 k ] T = [ k − 2 k ] = k [ 1 − 2 ] : k ∈ ℝ Here, k is a scalar. The eigenvector corresponding to the eigenvalue λ 2 = − 4 is determined based on the linear system given as, ( A − λ I ) v = 0 ( [ − 12 − 7 16 10 ] − [ λ 0 0 λ ] ) [ v 1 v 2 ] = [ 0 0 ] [ − 12 − λ − 7 16 10 − λ ] [ v 1 v 2 ] = [ 0 0 ] Substitute − 4 for λ in the above equation

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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 7.3, Problem 21P

To determine

The solution of the given system of differential equations based on the idea introduced in this section.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.3, Problem 20P

Chapter 7.3, Problem 22P

Students have asked these similar questions

Use the above to solve the following system of differential equations: 31= 1+2x2 3/2 = 2x1 + x₂

Answer no 4

Solve the following differential equations.

Chapter 7 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 7.1 - Prob. 1P Ch. 7.1 - Prob. 2P Ch. 7.1 - Prob. 3P Ch. 7.1 - Prob. 4P Ch. 7.1 - Prob. 5P Ch. 7.1 - Given that v1=(2,1) and v2=(1,1) are eigenvectors...Ch. 7.1 - Prob. 7P Ch. 7.1 - Prob. 8P Ch. 7.1 - Prob. 9P Ch. 7.1 - Prob. 11P

Knowledge Booster

$Algebra$

Learn more about

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

The solution of the given system of differential equations based on the idea introduced in this section.

Solution Summary: The author explains the solution of the given system of differential equations based on the idea introduced in this section.

Concept explainers

Videos

The solution of the given system of differential equations based on the idea introduced in this section.

Want to see the full answer?

Chapter 7 Solutions