# Second order DE as a system of first order DEs, find eigenvalues and critical points

1. Aug 10, 2011

### Rubik

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

Express y'' + 5y' - 24y = 0 as a system of couple first order DEs, find the eigenvalues of the system and the nature of the critical point at the origin. As well as find the general solution to the system of coupled equations and sketch some trajectories in the phase plane.

2. Relevant equations

3. The attempt at a solution
I have no idea really where to start so this could be completely wrong..

Using y1 = y and y2 = y''
and (A-lambda I)x = 0

0 = $\lambda$2 + 5$\lambda$ - 24
0 = ($\lambda$+8)($\lambda$-3)
$\lambda$1 = -8, $\lambda2$ = 3

From $\lambda$1 = -8;
x(1) = [1; -8] (vector)

And from $\lambda$2 = 3;
x(2) = [1; 3]

So my general solution is y(t) = $\alpha$x(1)e-8t + $\beta$x(2)e3t

And from there I am stuck and not even sure if that is on the right path, let alone how to determine the nature of the critical point at the origin.. any ideas?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Aug 10, 2011

### Rubik

Re: Second order DE as a system of first order DEs, find eigenvalues and critical poi

Then using those results because I have two real eigenvalues of opposite signs does this mean with the nature of the critical point at the origin it is in fact a saddle point?