Okay, I know that this is probably a simple question but I've always been good at doing the complicated things and bad at doing the easy things :D(adsbygoogle = window.adsbygoogle || []).push({});

Here's what I've got:

Find the general solution for the system of coupled ODEs. Determine kind and stability of the critical point. Sketch phase portrait.

Y'1 = -3Y1 - 2Y2

Y'2 = -2Y1 - 3Y2

I constructed a system of matrices and determined the characteristics of the systems as follows:

(sorry I gave up on trying to construct the matrix with TeX, hope this still makes sense)

p = -6

q = 5

[tex]\Delta[/tex] = 16

[tex]\lambda[/tex]1 = -1

[tex]\lambda[/tex]2 = -5

I then applied the two values for [tex]\lambda[/tex] to the original matrix and found two vector solutions for the system as follows:

x[tex]^{1}[/tex] = (1 -1)[tex]^{T}[/tex]

x[tex]^{2}[/tex] = (1 1)[tex]^{T}[/tex]

So the general solutions are:

x[tex]^{1}[/tex]e[tex]^{-t}[/tex]

x[tex]^{2}[/tex]e[tex]^{-5t}[/tex]

General Solution to the system:

Y(t) = c1x1e^-t + c2x2e^-5t

(sorry for the messy equation, TeX decided it didn't like me :D)

I also determined the characteristics of the node as being a stable, attractive improper node.

Now I just have to sketch the phase plane and I have NO idea where to start with this. Its probably stupidly easy and I'm just being daft, but I don't know what I'm supposed to do next. Any help would be fantastic. Thanks.

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# Need help drawing phase portraits for coupled systems of ODEs

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