# Second-Order Equations and Eigenvectors

1. Nov 30, 2011

### tatianaiistb

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 30, 2011

### 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?

3. Nov 30, 2011

### 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. :-(

4. Nov 30, 2011

### I like Serena

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

Are they related?

5. Nov 30, 2011

### tatianaiistb

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

6. Nov 30, 2011

### I like Serena

Ok.
(Did you find your relevant equation?)