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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.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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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/dt (x(t))
           y(t)) =A (x(t)
                             y(t)) 
 2. Find a change of co-ordinates, corresponding to theeigenvectors 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.
Transcribed Image Text: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.
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