Toronto Math Forum
MAT2442014F => MAT244 MathLectures => Topic started by: Bruce Wu on December 05, 2014, 03:23:08 PM

What is the difference between integrable and nonintegrable 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?

2x2ystem is integrable if there exists nontrivial (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 cyclesâ€”which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is nondegenerate).

But from H(x,y) = c, how do we determine the directions of trajectories?

Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cyclesâ€”which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is nondegenerate).
Also, all diagonal linear 2x2 systems are integrable, but those can be nodes, how is that explained?

Also, all diagonal linear 2x2 systems are integrable,
Wrong!! You confuse explicit solution with integrability. See definition

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.

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

Thanks