Solving a System of Coupled DEs: Eigenvalues & Trajectories

[Rubik]

Homework Statement

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.

Homework Equations

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 = \lambda² + 5\lambda - 24
0 = (\lambda+8)(\lambda-3)
\lambda₁ = -8, \lambda2 = 3

From \lambda₁ = -8;
x⁽¹⁾ = [1; -8] (vector)

And from \lambda₂ = 3;
x⁽²⁾ = [1; 3]

So my general solution is y(t) = \alphax⁽¹⁾e^-8t + \betax⁽²⁾e^3t

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?

 

[Rubik]

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?