Problem 1. Consider each of the following systems of ODES (where x1 and 2 are both dependent variables of the inde- pendent variable t. For each system below: • Write each system in vector/matrix form. • Find the eigenvectors and eigenvalues of the system by hand. • Use these to construct a real-valued general solution. Then use the initial conditions to find the particular solution. • Graph a component plot (plotting x₁ (t) and x2(t) against t, showing how each component evolves with time). • Graph a phase portrait of the system (on the x1-x2 axis), and use directional arrows to show how the solution evolves as t increases. (c) Sx1'(t) = 2x1(t) + 4x2(t) when [x2' (t) = −x1(t)+2x2(t) √x1(0) = [x2(0) = 1 1 (d) [x₁'(t) = -x1(t) + x2(t) √x1(0) = 3 when \x2' (t) = −2x1(t) + x2(t) ' |x2(0) = 0
Problem 1. Consider each of the following systems of ODES (where x1 and 2 are both dependent variables of the inde- pendent variable t. For each system below: • Write each system in vector/matrix form. • Find the eigenvectors and eigenvalues of the system by hand. • Use these to construct a real-valued general solution. Then use the initial conditions to find the particular solution. • Graph a component plot (plotting x₁ (t) and x2(t) against t, showing how each component evolves with time). • Graph a phase portrait of the system (on the x1-x2 axis), and use directional arrows to show how the solution evolves as t increases. (c) Sx1'(t) = 2x1(t) + 4x2(t) when [x2' (t) = −x1(t)+2x2(t) √x1(0) = [x2(0) = 1 1 (d) [x₁'(t) = -x1(t) + x2(t) √x1(0) = 3 when \x2' (t) = −2x1(t) + x2(t) ' |x2(0) = 0
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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Transcribed Image Text:Problem 1. Consider each of the following systems of ODES (where x1 and 2 are both dependent variables of the inde-
pendent variable t. For each system below:
• Write each system in vector/matrix form.
• Find the eigenvectors and eigenvalues of the system by hand.
• Use these to construct a real-valued general solution. Then use the initial conditions to find the particular
solution.
• Graph a component plot (plotting x₁ (t) and x2(t) against t, showing how each component evolves with time).
• Graph a phase portrait of the system (on the x1-x2 axis), and use directional arrows to show how the solution
evolves as t increases.
(c)
Sx1'(t) = 2x1(t) + 4x2(t)
when
[x2' (t) = −x1(t)+2x2(t)
√x1(0) =
[x2(0) = 1
1
(d)
[x₁'(t) = -x1(t) + x2(t)
√x1(0) = 3
when
\x2' (t) = −2x1(t) + x2(t)
'
|x2(0) = 0
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