roblem 1. (Based on Brennan Boyce 2e $3.4 #7) Consider the 2 x 2 system of ODEs, where X (t) = (x1(t) for some real-valued functions x1(t) and x2(t). X'(1) = () x 4 X(0) = X(t). Do the following: 1. Find the eigenvalues and eigenvectors of A. (Note that these will be complex!). 2. Find the general solution (ignoring initial condition) with real components. 3. Find the particular solution (using the initial condition).

roblem 1. (Based on Brennan Boyce 2e $3.4 #7) Consider the 2 x 2 system of ODEs, where X (t) = (x1(t) for some real-valued functions x1(t) and x2(t). X'(1) = () x 4 X(0) = X(t). Do the following: 1. Find the eigenvalues and eigenvectors of A. (Note that these will be complex!). 2. Find the general solution (ignoring initial condition) with real components. 3. Find the particular solution (using the initial condition).

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 1. (Based on Brennan Boyce 2e $3.4 #7)
()
Consider the 2 x 2 system of ODES, where X(t) =
for some real-valued functions x1(t) and x2(t).
4
X(0) =
3
X(t)
Do the following:
1. Find the eigenvalues and eigenvectors of A. (Note that these will be complex!).
2. Find the general solution (ignoring initial condition) with real components.
3. Find the particular solution (using the initial condition).
4. Plot each component of the solution as a function of time.
5. Plot the vector plot (i.e. on the
X1X2-plane).
6. Using your particular solution, approximate the values of x, (t) and x2(t)
(a) when t =
(b) when t =2
7. What is the maximum value of x, in the interval 0 <t< 2. (You can use a graph). When does it occur?
8. What is the maximum value of x, in the same interval? When does it occur?

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