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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 = [itex]\lambda[/itex]2 + 5[itex]\lambda[/itex] - 24
0 = ([itex]\lambda[/itex]+8)([itex]\lambda[/itex]-3)
[itex]\lambda[/itex]1 = -8, [itex]\lambda2[/itex] = 3
From [itex]\lambda[/itex]1 = -8;
x(1) = [1; -8] (vector)
And from [itex]\lambda[/itex]2 = 3;
x(2) = [1; 3]
So my general solution is y(t) = [itex]\alpha[/itex]x(1)e-8t + [itex]\beta[/itex]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?