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For a linear system in 2 dimensions, dx dt = x(t) Ax, where x = y(t) and A is a 2 × 2 constant matrix, show that x(t) = etA. is the solution, with x(0) : = x0. (2) Xo, where x0 = (xo, Yo), Write the linear system dx dy = 2x-3y, = 4x-5y dt dt in matrix form (2), and find the eigenvalues and eigenvectors. Deduce the type and stability of the fixed point at (0,0) and sketch the phase portrait of (3).

For a linear system in 2 dimensions, dx dt = x(t) Ax, where x = y(t) and A is a 2 × 2 constant matrix, show that x(t) = etA. is the solution, with x(0) : = x0. (2) Xo, where x0 = (xo, Yo), Write the linear system dx dy = 2x-3y, = 4x-5y dt dt in matrix form (2), and find the eigenvalues and eigenvectors. Deduce the type and stability of the fixed point at (0,0) and sketch the phase portrait of (3).

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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For a linear system in 2 dimensions, dx dt = x(t) Ax, where x = y(t) and A is a 2 × 2 constant matrix, show that x(t) = etA. is the solution, with x(0) : = x0. (2) Xo, where x0 = (xo, Yo), Write the linear system dx dy = 2x-3y, = 4x-5y dt dt in matrix form (2), and find the eigenvalues and eigenvectors. Deduce the type and stability of the fixed point at (0,0) and sketch the phase portrait of (3).
Transcribed Image Text:For a linear system in 2 dimensions, dx dt = x(t) Ax, where x = y(t) and A is a 2 × 2 constant matrix, show that x(t) = etA. is the solution, with x(0) : = x0. (2) Xo, where x0 = (xo, Yo), Write the linear system dx dy = 2x-3y, = 4x-5y dt dt in matrix form (2), and find the eigenvalues and eigenvectors. Deduce the type and stability of the fixed point at (0,0) and sketch the phase portrait of (3).
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