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What type of fixed point is the equilibrium solution x = y = 0? Sketch the phase portrait of the linear system. Find the solution of the linear system corresponding to the initial conditions x(0) = 1, y(0) = 0. Determine the values limt→∞ x(t) and limt→∞ y(t).

What type of fixed point is the equilibrium solution x = y = 0? Sketch the phase portrait of the linear system. Find the solution of the linear system corresponding to the initial conditions x(0) = 1, y(0) = 0. Determine the values limt→∞ x(t) and limt→∞ y(t).

Advanced Engineering Mathematics
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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Hi. Please help me with the solutions to ONLY PART C AND D. I got the last two parts to this question wrong

Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: x = 2 tanh(x) — 2x cos(y) + ex+³y – 1, ý = 1 3 cosh (x) - 3ey + 24 + sin(x). (a) Write down in matrix form of the type X = AX with X = (ry) 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→∞ x(t) and limt→∞ y(t).
Transcribed Image Text:Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: x = 2 tanh(x) — 2x cos(y) + ex+³y – 1, ý = 1 3 cosh (x) - 3ey + 24 + sin(x). (a) Write down in matrix form of the type X = AX with X = (ry) 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→∞ x(t) and limt→∞ y(t).
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