Problem 2. (Based on Brennan Boyce 2e $3.5 #9)X1(t)Consider the 2 x 2 system of ODES, where X (t) =for some real-valued functions x1(t) and x2(t).X2(t)X'(t) =X (t)X (0)Do the following:1. Find the eigenvalues and eigenvectors of A. (Note that these will be repeated!).2. Find a second linearly independent solution.3. Directly show that your two solutions are linearly independent for all values of t.4. Find the general solution (ignoring initial condition), which is a linear combination of these two linearly inde-pendent solutions.5. Find the particular solution (using the initial condition).6. Plot each component of the solution as a function of time.7. Plot the vector plot (i.e. on the x1x2-plane).

Answered: Image /qna-images/answer/07c93031-c1b4-4de4-9535-93d12f974ed7.jpg

Problem 2. (Based on Brennan Boyce 2e $3.5 #9) (x1(t) Consider the 2 x 2 system of ODES, where X (t) = for some real-valued functions x1(t) and x2(t). X2(t) X'(t) = X (t) = (0) X Do the following: 1. Find the eigenvalues and eigenvectors of A. (Note that these will be repeated!).

Problem 2. (Based on Brennan Boyce 2e $3.5 #9) (x1(t) Consider the 2 x 2 system of ODES, where X (t) = for some real-valued functions x1(t) and x2(t). X2(t) X'(t) = X (t) = (0) X Do the following: 1. Find the eigenvalues and eigenvectors of A. (Note that these will be repeated!).

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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Question

Problem 2. (Based on Brennan Boyce 2e $3.5 #9)
X1(t)
Consider the 2 x 2 system of ODES, where X (t) =
for some real-valued functions x1(t) and x2(t).
X2(t)
X'(t) =
X (t)
X (0)
Do the following:
1. Find the eigenvalues and eigenvectors of A. (Note that these will be repeated!).
2. Find a second linearly independent solution.
3. Directly show that your two solutions are linearly independent for all values of t.
4. Find the general solution (ignoring initial condition), which is a linear combination of these two linearly inde-
pendent solutions.
5. Find the particular solution (using the initial condition).
6. Plot each component of the solution as a function of time.
7. Plot the vector plot (i.e. on the x1x2-plane).

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ISBN:

9780470458365

Author:

Erwin Kreyszig

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Wiley, John & Sons, Incorporated