1. Find a matrix A such that the linear system d dt x(t) = −7x+2y d dt y(t) = −1x-8y is given by d x(t) dt y(t) x(t) A y(t) 2. Find a change of co-ordinates, corresponding to the eigenvectors of A, such that the differential equation is un-coupled in the new co-ordinates. 3. Calculate X(t) = exp(At) and show this matrix satisfies the linear differential equation. 4. Characterise the trivial equilibrium and sketch the phase portrait in the new coor- dinates.

1. Find a matrix A such that the linear system d dt x(t) = −7x+2y d dt y(t) = −1x-8y is given by d x(t) dt y(t) x(t) A y(t) 2. Find a change of co-ordinates, corresponding to the eigenvectors of A, such that the differential equation is un-coupled in the new co-ordinates. 3. Calculate X(t) = exp(At) and show this matrix satisfies the linear differential equation. 4. Characterise the trivial equilibrium and sketch the phase portrait in the new coor- dinates.

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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