2. Suppose you analyze a 2D linear system x = Ax and find complex eigenvalues λ = λr ±iλi where X₂ = 7/2 and λ; = √√/A − (7/2)², both real. These have associated eigenvectors V+, and the the general solution is x(t) = eλrt (C+eïdi¹v+ +C_e¯ï¾‹ºv_). Use Euler's relation ei = cos 0 + i sin 0, and the fact that v₁ = V₁ ±iv, where vr and vi are both real column vectors with two entries (* shown in notes or in recitation) to write the solution in the form x(t): = : eλrt [B₁ (vr cos(λ¡t) — vį sin(\įt)) + B2 (vr sin(\¿t) + vį cos(\¿t))] How are B₁, B2 related to C+, C_? Show that even though you may find an i in one of the expressions, if the initial condition is real, then the solution will be real for all t.
2. Suppose you analyze a 2D linear system x = Ax and find complex eigenvalues λ = λr ±iλi where X₂ = 7/2 and λ; = √√/A − (7/2)², both real. These have associated eigenvectors V+, and the the general solution is x(t) = eλrt (C+eïdi¹v+ +C_e¯ï¾‹ºv_). Use Euler's relation ei = cos 0 + i sin 0, and the fact that v₁ = V₁ ±iv, where vr and vi are both real column vectors with two entries (* shown in notes or in recitation) to write the solution in the form x(t): = : eλrt [B₁ (vr cos(λ¡t) — vį sin(\įt)) + B2 (vr sin(\¿t) + vį cos(\¿t))] How are B₁, B2 related to C+, C_? Show that even though you may find an i in one of the expressions, if the initial condition is real, then the solution will be real for all t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![2. Suppose you analyze a 2D linear system ẋ = Ax and find complex eigenvalues
\[
\lambda_{\pm} = \lambda_r \pm i\lambda_i
\]
where \(\lambda_r = \tau / 2\) and \(\lambda_i = \sqrt{\Delta - (\tau/2)^2}\), both real. These have associated eigenvectors \(\mathbf{v}_{\pm}\), and the general solution is
\[
\mathbf{x}(t) = e^{\lambda_r t} \left( C_+ e^{i \lambda_i t} \mathbf{v}_+ + C_- e^{-i \lambda_i t} \mathbf{v}_- \right).
\]
Use Euler’s relation \(e^{i\theta} = \cos \theta + i \sin \theta\), and the fact that \(\mathbf{v}_{\pm} = \mathbf{v}_r \pm i \mathbf{v}_i\) where \(\mathbf{v}_r\) and \(\mathbf{v}_i\) are both real column vectors with two entries (* shown in notes or in recitation) to write the solution in the form
\[
\mathbf{x}(t) = e^{\lambda_r t} \left[B_1 \left(\mathbf{v}_r \cos(\lambda_i t) - \mathbf{v}_i \sin(\lambda_i t)\right) + B_2 \left(\mathbf{v}_r \sin(\lambda_i t) + \mathbf{v}_i \cos(\lambda_i t)\right)\right]
\]
How are \(B_1, B_2\) related to \(C_+, C_-\)? Show that even though you may find an \(i\) in one of the expressions, if the initial condition is real, then the solution will be real for all \(t\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2F43004ee1-95e9-4d70-82de-39f441cc18cc%2F60oeew_processed.png&w=3840&q=75)
Transcribed Image Text:2. Suppose you analyze a 2D linear system ẋ = Ax and find complex eigenvalues
\[
\lambda_{\pm} = \lambda_r \pm i\lambda_i
\]
where \(\lambda_r = \tau / 2\) and \(\lambda_i = \sqrt{\Delta - (\tau/2)^2}\), both real. These have associated eigenvectors \(\mathbf{v}_{\pm}\), and the general solution is
\[
\mathbf{x}(t) = e^{\lambda_r t} \left( C_+ e^{i \lambda_i t} \mathbf{v}_+ + C_- e^{-i \lambda_i t} \mathbf{v}_- \right).
\]
Use Euler’s relation \(e^{i\theta} = \cos \theta + i \sin \theta\), and the fact that \(\mathbf{v}_{\pm} = \mathbf{v}_r \pm i \mathbf{v}_i\) where \(\mathbf{v}_r\) and \(\mathbf{v}_i\) are both real column vectors with two entries (* shown in notes or in recitation) to write the solution in the form
\[
\mathbf{x}(t) = e^{\lambda_r t} \left[B_1 \left(\mathbf{v}_r \cos(\lambda_i t) - \mathbf{v}_i \sin(\lambda_i t)\right) + B_2 \left(\mathbf{v}_r \sin(\lambda_i t) + \mathbf{v}_i \cos(\lambda_i t)\right)\right]
\]
How are \(B_1, B_2\) related to \(C_+, C_-\)? Show that even though you may find an \(i\) in one of the expressions, if the initial condition is real, then the solution will be real for all \(t\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
