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
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
Related questions
Question
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 1 steps
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning