Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: 1 1 ż = 2 tanh(z) — 2x cos(y) + e²+3y = 1, y = 3 cosh(r) - 3e² + y + sin(x). (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 z = 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. (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 lim(t) and lim, y(t). -00
Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: 1 1 ż = 2 tanh(z) — 2x cos(y) + e²+3y = 1, y = 3 cosh(r) - 3e² + y + sin(x). (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 z = 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. (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 lim(t) and lim, y(t). -00
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider a system of two nonlinear first-order ODEs, where x and y are functions of the
independent variable t:
1
1
* = 2 tanh(x) — 2x cos(y) +²+³y-1, y = 3 cosh(x) - 3e+y+sin(x).
(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.
(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 limt (t) and lim+→∞ y(t).
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