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January 4th, 2019, 10:25 AM   #1
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Stability of differential equation

Hi, I have big problem with solving one of exercises in my analysis course and I will be glad for any help.
I have ordinary differential equation:


and I have to examine stability of the above system.
I don't know even how to start - could someone explain me what the matrix means (normally I had equation like y'(x) + y(x) + 5x = 7 - maybe harder, but in one line, I see matrix first time) and how to start with this problem (examining stability)?

Last edited by skipjack; January 6th, 2019 at 02:49 AM.
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January 4th, 2019, 12:51 PM   #2
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You have two simultaneous equations. $Dy_1=-2y_1+3y_2+x^{-2}$ and $Dy_2=3y_1+y_2+x^{-2}$.
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January 5th, 2019, 06:53 AM   #3
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This is just a standard linear differential question. Generically, these are equations of the form'
\[y' = A(x)y + F(x)\]
where $A$ is a matrix and $y, F$ are vectors. This should be covered in a first course on differential equations. If not, then you really can't even parse the question much less answer it.

Now, your question as stated doesn't make sense. Differential equations don't have a stability. Stability is a property possessed by invariant sets which are unions of trajectories for a differential equation. Most commonly, you look at a single trajectory such as an equilibrium or a periodic solution. You need to rephrase your question before anyone can understand how to answer it.
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January 5th, 2019, 08:17 AM   #4
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Thanks a lot for an answer. The problem is I can't add much more, I have just said:

Consider the system of the ordinary differential equations:

--here exactly what is on the picture--

Examine the stability of the above system.

Last edited by skipjack; January 6th, 2019 at 02:45 AM.
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