Lecture 0201 question

[Suheng Yao]

This is the last question from yesterday's lecture 0201 section. I still don't understand why there is a negative sign on the right side of the equation?

[Victor Ivrii]

Indeed, there should be no $-$ on the right, unless I change $b-y$ to $y-b$ (which I intended to to but doid not). I updated handout, it is just a single place as on the next frame everything is right.

[Suheng Yao]

Thanks, prof. Also, on the next slide, I feel confused about why does f(x) have a single minimum at x=a and have equilibrium at x=a and y=b? I really don't get the idea here.

[Victor Ivrii]

You can investigate $f(x)$ as in Calculus I.

Also because $x=x(t)$ and $y=y(t)$ and $x=a,y=b$ is a constant solution (equilibrium). Look at the picture on the next slide. We excluded $t$ from our analysis but it does not mean that it had gone

[Jianfeng Huang]

I was also wondering how to extent the concept of direction field in this case, as both x and y are functions of t. Why the direction field shows the relationship between x and y? Is it because equations do not include t explicitly, and theoretically we can have 3-dimensional direction field?

[Victor Ivrii]

We use $x$ and $y$ because we can exclude $t$ but neither $x$ nor $y$