Directional field

[Nicole Frazer]

Hello,

I wanted to clear up any confusion about which homework problems we should be practicing as of where we are in course right now. Should we be doing the Chapter 1 Questions  this week, for sections 1.1, 1.2, and 1.3 posted on Quercus? The topics of directional fields have me a bit confused.

Thank you, I appreciate any help provided!

[Victor Ivrii]

Basically, directional fields as in Chapter 1 are more or less "free hand sketch" and a lot of heuristics, speculations, and imagination.

In Chapter 9 we will come basically to this, and then we will really exploit integral curves and "phase portraits"

On Outlines page, section "Learning Resources" we have ODE Plotters (remember, the black triangle is expandable). Go f.e. http://www.bluffton.edu/homepages/facstaff/nesterd/java/slopefields.html, click "System" tab and enter data, next, click on some point on the picture, an voilà, you have a curve. Do it again, and again, ... It is kinda fun
 

[Shujia Wu]

Take the q24 and q23(ch.1) in textbook as examples, how to draw direction fields that involves $e^t$ and $\sin(t)$? Still just calculate the value of $y'$ for each $t$ andy?



23. $y′ = e^{−t} + y$
24. $y′ = 3 \sin t + 1 + y$

[Victor Ivrii]

Don't calculate. Just consider general behavior of these functions. And check using online plotter I mentioned.

[Chengmin Jiang]

For this course(or for tests), which one is important, draw a perfect direction field or understand the behavior of the function or even both?

[Kathryn Bucci]

I don't think it would be possible to draw a perfect direction field and if you understand the behaviour of the function then you should be able to produce an adequate direction field.

[Celine Park]

Hello, I'm confused with drawing direction fields.

In class, I learned that the slope for the same $y$ value will be equal for any time $t$.
However, while reviewing and studying with outside resources, I found direction fields drawn without having the same slope for the same $y$ value.
Why and in which cases are they drawn differently?

Could you make this clear for me?
Thank you.

[Tzu-Ching Yen]

When your equation for $y'$ only has dependence on $y$, directional field has same value for same $y$.

But $y'$ could have dependence on other variables like $t$. $y' = y + t$ for example, even with $y$ fixed, could yield different slope at different $t$.