Problem 1. Consider each of the following systems of ODES (where x₁ and x2 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. √x1(0) = −1 x2(0) = 2 (a) Sx1'(t) = 2x1(t) + 2x2(t) when \x2' (t) = 6x1(t) + 3x2(t) ' (b) [x1'(t) = x1(t) + x2(t) when √x₁(0) = = 3 \x2' (t) = 2x1(t) + 2x2(t) |x2(0) = 0
Problem 1. Consider each of the following systems of ODES (where x₁ and x2 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. √x1(0) = −1 x2(0) = 2 (a) Sx1'(t) = 2x1(t) + 2x2(t) when \x2' (t) = 6x1(t) + 3x2(t) ' (b) [x1'(t) = x1(t) + x2(t) when √x₁(0) = = 3 \x2' (t) = 2x1(t) + 2x2(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 x2 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.
√x1(0) = −1
x2(0) = 2
(a)
Sx1'(t) = 2x1(t) + 2x2(t)
when
\x2' (t) = 6x1(t) + 3x2(t)
'
(b)
[x1'(t) = x1(t) + x2(t)
when
√x₁(0) =
= 3
\x2' (t) = 2x1(t) + 2x2(t)
|x2(0) = 0
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