Second-Order Equations and Eigenvectors

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Homework Help Overview

The problem involves converting a second-order differential equation, specifically y"=0, into a first-order system represented as du/dt=Au. The discussion centers around the matrix A derived from the system and its properties, including eigenvalues and eigenvectors, as well as computing the matrix exponential eAt.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to compute eAt and expresses confusion regarding how to apply it to the initial conditions given. Some participants question the definition of u and its relationship to the system, while others inquire about relevant equations for solving linear systems of first-order differential equations.

Discussion Status

The discussion is ongoing, with some participants providing clarifications and asking probing questions. The original poster indicates progress in understanding but has not confirmed a complete resolution to their confusion.

Contextual Notes

The original poster notes a lack of information beyond the problem statement, which may impact their ability to proceed. There is also a mention of uncertainty regarding the relationship between the variables defined in the problem.

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Homework Statement



Convert y"=0 to a first-order system du/dt=Au

d/dt [y y']T = [y' 0]T = [0 1; 0 0] [y y']T

This 2x2 matrix A has only one eigenvector and cannot be diagonalized. Compute eAt from the series I+At+... and write the solution eAtu(0) starting from y(0)=3, y'(0)=4. Check that your (y, y') satisfies y"=0.

Homework Equations





The Attempt at a Solution



So I found eAt to be equal to the matrix
[1 t
0 1].
I found this eAt=I+At where A is the matrix [0 1; 0 0].

I also know that matrix A has eigenvalue 0 with multiplicity 2, and eigenvector [1 0]T.

But from there I'm stuck... Not sure how to get eAtu(0)...

Can anyone help? Thanks in advance!
 
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Hi again!

How is u defined in your problem?

And do you have a relevant equation for solving a linear system of first order differential equations?
 
That's why I'm so confused... The only information given is the one I stated above exactly as it is worded. And I don't know of any relevant equations for solving this system. :-(
 
In your problem statement you write du/dt.
And then you write d/dt [y y']^T.

Are they related?
 
Nevermind. I think I figured it out. Thanks for the help!
 
Ok.
(Did you find your relevant equation?)
 

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