I Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: 1 1 ở = 2 tanh(x) − 2x cos(y) + e²+³y = 1, y = 3 cosh(x) − 3e² + y + sin(x). - (a) Write down in matrix form of the type X = AX with X = (x, y)T the system obtained by linearisation of the above equations around the point x = y = 0. Specify the elements of the matrix A. (b) Find the eigenvalues and eigenvectors of the matrix A obtained in (a). Write down the general solution of the linear system. I (c) What type of fixed point is the equilibrium solution x=y=0? Sketch the phase portrait of the linear system. the initial conditions.
I Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: 1 1 ở = 2 tanh(x) − 2x cos(y) + e²+³y = 1, y = 3 cosh(x) − 3e² + y + sin(x). - (a) Write down in matrix form of the type X = AX with X = (x, y)T the system obtained by linearisation of the above equations around the point x = y = 0. Specify the elements of the matrix A. (b) Find the eigenvalues and eigenvectors of the matrix A obtained in (a). Write down the general solution of the linear system. I (c) What type of fixed point is the equilibrium solution x=y=0? Sketch the phase portrait of the linear system. the initial conditions.
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
Solutions please
Transcribed Image Text:Consider a system of two nonlinear first-order ODEs, where x and y are functions of the
independent variable t:
* = 2 tanh(x) − 2x cos(y) + e*+ 3y - 1, y = 3 cosh(x) - 3e+y+sin(x).
1
1
(a) Write down in matrix form of the type X = AX with X = (x, y) the system
obtained by linearisation of the above equations around the point x = y = 0.
Specify the elements of the matrix A.
(b) Find the eigenvalues and eigenvectors of the matrix A obtained in (a). Write
down the general solution of the linear system.
I
(c) What type of fixed point is the equilibrium solution x=y=0? Sketch the phase
portrait of the linear system.
(d) Find the solution of the linear system corresponding to the initial conditions
x(0) = 1, y(0) = 0. Determine the values limto r(t) and limto y(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 4 steps with 3 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,
