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Classify the fixed point at the origin and sketch an accurate phase portrait for the following system: $$\left\{\begin{matrix} \dfrac{dx}{dt}=36x-16y\\ \dfrac{dy}{dx}=-3x+28y \end{matrix}\right.$$

Am I correct in thinking that I need to write these two equations in matrix form, find the eigenvalues and depending on what they are will determine what fixed point I have? To sketch the phase portrait do I then need to know the eigenvectors and directions?

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    Welcome to MSE. Are you sure that the second equation is $dy/dx$ and not $dy/dt$?2012-08-12
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    That's what I'm confused about! The question is definitely dy/dx. If I had dy/dt instead I would have no problem doing this question.2012-08-12
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    It is probably a misprint. Otherwise, you'd end up with a nonlinear system.2012-08-12
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    If I take it as dy/dt instead of dy/dx and find the eigenvalues and eigenvectors do I need to know the eigendirections to draw the phase portrait? How would I go about finding the eigendirection?2012-08-12
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    The eigendirections are *given* by the eigenvectors. Put differently, all the eigenvectors for a given eigenvalue form a linear space (at least if you add the zero vector, which is technically not an eigenvector, but still satisfies the defining equation of eigenvectors), and you can think of this space, which is typically one-dimensional, as specifying a direction. And that direction is the eigendirection.2012-08-12

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