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Class05 Online

The document provides extra questions and partial answers from a class on differential equations. The questions ask about sketching phase portraits for different 1-dimensional ODEs and finding ODEs with certain properties related to stationary points. Hints and possible answers are given for each question.

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12 views2 pages

Class05 Online

The document provides extra questions and partial answers from a class on differential equations. The questions ask about sketching phase portraits for different 1-dimensional ODEs and finding ODEs with certain properties related to stationary points. Hints and possible answers are given for each question.

Uploaded by

Jason
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Differential Equations 2019/20

MA 209

Extra questions from class 5, with partial answers

Question
Suppose the 1-dimensional ODE x0 = f (x) has the following phase portrait:
P
 t P
 t 
P t P

0 x -

(a) Sketch the phase portrait of the ODE x0 = xf (x).


(b) Sketch the phase portrait of the ODE x0 = x2 f (x).
(c) Find a 1-dimensional ODE x0 = g(x) that has an infinite number of stationary points,
and where all stationary points are attractors or repellors.
(d) Find a 1-dimensional ODE x0 = h(x) that has an infinite number of stationary points,
and where all stationary points are shunts.
(e) Find a 1-dimensional ODE x0 = j(x) that has two stationary points, and where both
stationary points are attractors.

Answers/hints:
(a) We have an additional stationary point at x = 0. For x > 0, the sign of xf (x) is the
same as the sign of f (x); for x < 0, the sign of xf (x) is the opposite of the sign of f (x).
So we get the following phase portrait:

P t 
P t P
 t 
P t P

0 x -

(b) We have an additional stationary point at x = 0. But now for all x 6= 0, the sign of
x2 f (x) is the same as the sign of f (x). Hence we get the following phase portrait:
P
 t P
 t P
 t 
P t P

0 x -

(c) Many options are possible. The most important observation is that we need a function
f (x) with an infinite number of x for which f (x) = 0, and then make sure it has the right
properties. One option is f (x) = sin(x). Hence taking the ODE x0 = sin(x).

c London School of Economics, 2019


MA 209 Differential Equations Extra questions from class 5, with partial answers — Page 2

(d) Many options are again possible. The most important observation is that we need a
function f (x) with an infinite number of x for which f (x) = 0, and then make sure it
has the right properties. In particular we must make sure that the sign of f (x) is the
same for all non-stationary points. Possibilities are x0 = |sin(x)|, or x0 = sin2 (x), or
x0 = sin(x) + 1.
(e) This is a trick question: such an ODE is not possible. If you try to make a phase portrait
with this property, it is easy to see that you just can’t make it.

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