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# Differential Equation Phase Plane Portrait. watch

1. So given a non homogenous system of ode:

x'=-x+6y -2
y'=-x+4y

sketch the phase portrait.

So I first calculated the eigen values/vectors of the matrix
|-1 6|
|-1 4|
and find for lambda=2, eigen vector = (2,1)
lambda=1, eigen vector = (3,1)

and particular solution = (4,1)

How do I sketch the phase portrait?
2. (Original post by specimenz)
So given a non homogenous system of ode:

x'=-x+6y -2
y'=-x+4y

sketch the phase portrait.

So I first calculated the eigen values/vectors of the matrix
|-1 6|
|-1 4|
and find for lambda=2, eigen vector = (2,1)
lambda=1, eigen vector = (3,1)

and particular solution = (4,1)

How do I sketch the phase portrait?

Then note that you have for some constants

Now, as we have which means that every arrow will be coming from the critical point (so it's unstable), and furthermore, as therefore every vector will come out parallel to the eigenvector (your ).

Now consider , which means that and more specifically, they will both tend to infinity along the vector because at +ve infinity so that term takes control.

The rest is just sketching accordingly to this.

Small disclaimer: As I'm learning this topic at the moment, I haven't come across the case yet where or has a constant in its expression, but this is the analysis I'd expect

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