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Qualitative behaviour of a differential equation

I am unable to do Q8. The only experience i have of discussing the qualitative behaviour of DE's was in first year when there are a pair of de's, with dx/dt= and dy/dt= and you put them in matrix form. Thanks.

Q8(2).jpg
(edited 10 years ago)
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The sign of u' tells you whether u is decreasing or increasing when it takes a certain value.
Original post by matt2k8
The sign of u' tells you whether u is decreasing or increasing when it takes a certain value.


Yes but the solution talks about stable points etc
Answer 8.jpg
Reply 4
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BlueSam3
17
A fixed point of u is a point where u' = 0. With the format given, these are obvious.

Do you know how the stability of such points is related to the derivative of u' with respect to u at those points?
Reply 5
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DFranklin
18
Original post by BlueSam3
A fixed point of u is a point where u' = 0. With the format given, these are obvious.

Do you know how the stability of such points is related to the derivative of u' with respect to u at those points?
Note that you don't really need to consider the derivative of u'' here, it's sufficient to consider the sign of u' in the neighbourhood of each point.

[We did this at A-level, is it no longer taught?]
Reply 6
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BlueSam3
17
Original post by DFranklin
Note that you don't really need to consider the derivative of u'' here, it's sufficient to consider the sign of u' in the neighbourhood of each point.

[We did this at A-level, is it no longer taught?]


I think you've probably got your primes backwards there (the former should be u', no?), but yes.

And it's on some A-level courses, but not others for some reason.

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