Toronto Math Forum
MAT244-2014F => MAT244 Math--Lectures => Topic started by: Bruce Wu on December 05, 2014, 03:23:08 PM
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What is the difference between integrable and non-integrable systems of first order ODEs?
I remember the professor talking about it in class but I cannot find it in the textbook. I know what it means, but how does this property affect its solutions?
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2x2-ystem is integrable if there exists non-trivial (not identically constant) function $H(x,y)$ which is preserved along trajectories.
Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cycles—which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).
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But from H(x,y) = c, how do we determine the directions of trajectories?
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Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cycles—which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).
Also, all diagonal linear 2x2 systems are integrable, but those can be nodes, how is that explained?
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Also, all diagonal linear 2x2 systems are integrable,
Wrong!! You confuse explicit solution with integrability. See definition
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So please tell me if I am understanding it correctly now,
y = c*x^2 for all real numbers c represents a particular node. However, H(x,y) = y/x^2 = c is not preserved at x = 0, so it is not integrable, even though it has an explicit solution.
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However, $H(x,y) = y/x^2 = c$ is not preserved at $x = 0$, so it is not integrable, even though it has an explicit solution.
It is undefined at $(0,0)$ and cannot be defined as continuous function. But your feeling is correct
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Thanks