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Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: * =9 tanh(y) — 2c(e® − 1) — r® tanh(y), y = -2- (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. COS X X - 3(ev - 1)- y²e (b) Find the eigenvalues and eigenvectors of the matrix A find in (a). Write down the general solution of the linear system. (c) What type of fixed point is x=y=0?

Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: * =9 tanh(y) — 2c(e® − 1) — r® tanh(y), y = -2- (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. COS X X - 3(ev - 1)- y²e (b) Find the eigenvalues and eigenvectors of the matrix A find in (a). Write down the general solution of the linear system. (c) What type of fixed point is x=y=0?

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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Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: 76 3 * = 9 tanh(y) — 2x(e" - 1) - x³ tanh(y), y = -2. 1 - cos x -3(ey-1)-y²e T (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 = Specify the elements of the matrix A. =y=0. X (b) Find the eigenvalues and eigenvectors of the matrix A find in (a). Write down the general solution of the linear system. (c) What type of fixed point is x=y=0? (d) Find the solution of the linear system corresponding to the initial conditions x(0) = 2, y(0) = 0. Waco \I
Transcribed Image Text:Consider a system of two nonlinear first-order ODEs, where x and y are functions of the independent variable t: 76 3 * = 9 tanh(y) — 2x(e" - 1) - x³ tanh(y), y = -2. 1 - cos x -3(ey-1)-y²e T (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 = Specify the elements of the matrix A. =y=0. X (b) Find the eigenvalues and eigenvectors of the matrix A find in (a). Write down the general solution of the linear system. (c) What type of fixed point is x=y=0? (d) Find the solution of the linear system corresponding to the initial conditions x(0) = 2, y(0) = 0. Waco \I
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