Second-Order Equations and Eigenvectors

[tatianaiistb]

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 e^At from the series I+At+... and write the solution e^Atu(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 e^At to be equal to the matrix
[1 t
0 1].
I found this e^At=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 e^Atu(0)...

Can anyone help? Thanks in advance!

 

[I like Serena]

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?

 

[tatianaiistb]

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. :-(

 

[I like Serena]

In your problem statement you write du/dt.
And then you write d/dt [y y']^T.

Are they related?

 

[tatianaiistb]

Nevermind. I think I figured it out. Thanks for the help!

 

[I like Serena]

Ok.
(Did you find your relevant equation?)