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Numerade Educator



Problem 18 Hard Difficulty

Make a rough sketch of a direction field for the autonomous differential equation $ y' = f(y), $ where the graph of $ y $ is as shown. How does the limiting behavior of solutions depend on the value of $ y(0)? $


The first diagram makes sense only for autonomous first-order differential equations.
That means, the equations of the form $y^{\prime}=f(y)$ , where $y$ is a function of some variable $x($
The diagram of graphs is symmetric about the origin if the graph of $f(y)$ is symmetric about the $y$ axis, that means, $y$ is an even function, $f(-y)=f(y),$ as it seems to be.
Above figure indicate solution curves along with direction fields.
The direction fields are tangent to these curves.


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Video Transcript

Yeah. Grab wrong patient into determine running behavior of solution based on value. Very. The reason is was that one You The graph that ray function one. This is an art form equation. So me and one variable one when it wise some function, drop a soldier field those that I see several remember, is equal white. Fine. This is why so you see a little wine, All ex, but you're not. We can look at horizontal lines now. Alexey y axes. Now we see that you see it saying zero. We have one. You will at points 20 Yangon born zero one. Next people's so hi, this zero wise people to plus or minus one last night too. This is for all. So you see that along the lines White one on this one. U minus two. Chairman. What? And other We don't know for sure. The shops are which crosses. Why Axis wine. So si is equal to zero. Well, we haven't seasonable pepper. Why always zero people? One. There are a few different points. Have you seen now you zero one of which is greater than two. Must to Well, why? I want to, you know you hold Waas. It's a long Want some small remember and assume it's symmetric grab except Salon. Really plus or minus to Plus, did you see that? Econ zero is certainly a positive number. So how deposits along these? Where is on full on? But the this 100 isn't it has one. It could be super or oh, the man Post minus two sets to one again. Looks like one. Did you see? That is why it's close. It positive y increases. So you see that he's one fine still flying that between Want to? We see that in between Life off one, too. Have it wide. It seems to be which a truck he has. Maybe the direction I think this. And it's going to be hard seeing this graph. But no doubt between these two extremes Zeros preventive one God, sort of like that. This is also true for why, between made it to many of one tonight here, time slips between 01 zero Negative one. You see, that may be only one. Why increases from zero 21? Is that a zero Charlie of one? We had Michael's negative one Gradual increases slopes. I was here the approximation. I love you, they were asked. Have a living behavior the solution and on what value? For why? So this is essentially asking if we choose our initial condition to be somewhere on the Wyatt. What is the solution? Want to look right? Well, it having limits? Just a few examples. Black first garnish wise. You see 00 Well, white, one level. So the region as a way of life was one in the vending direction. Recent ill. It's pretty clear that yeah, negative one in certain less thing. Why you less than one you see from this graph In a minute, the exit perches positive A of Why that 21 hands as ex coaches infinity. Another test. Try the initial condition. Why is your own people to? But we could see clearly. We pick y zero equals one word, maybe of one. We have horizontal lines 0.1 like one, so we will have limits for those but revealing a marching case. Let's take zero to be 1.5. Isn't this crap in Great. You know how to train? Grab this. All right. Ideal pulls maybe 1.5. Also, we'll see that the first draft won't get 01 point by draft levels off like this one increases. That'll one. Likewise, the second graph the court zero, um, or grand decreases until levels off white too. Why? You see that? Yeah. To less. Why do you less negative? One or one is less than wise. You is less than two. And we have that. Let it expert, choose positive infinity on my next people to second case White too. Well, actually. Secondly, some more. The limits. Why not satisfies this condition? I like that. Want to born? You see that? Yeah. Wants us. Why not? Except purchase party unity? Why people to one? No, that's purchase. Meghan. Why is that people to again? We take why? To why do too wise too. We end up. I mean, two more horizontal line. Have. Then it's Tribune War has limit positive in a negative direction. And the other one has. So the last interesting case that we'll see is it choose? Why not? Or why not to be? Let's suppose that the absolute value of one not listen that could have one case. Why not? Well, like that. This in green and see my slip field. You have a point. Let's take some point. Really simple. Why not? People won't buy when you see that the occurrence quickly increase or expert increase until the care to Why people sort of next financial. So you see that limit excursions Infinity. Why? Why? Positive? You have a clinic that's proteins to like wise. There was less than men. Take for example why you demographic green have pointed zero, maybe two point by Can you see that the next approaches flattens out and ones here, inside by two experts creases quickly again The next often has been what did across the x axis. So you see that the next part going to why that people summarize X approaches. Why that people to to we have condition. Why zero? He also had been why zero is equal to this limit. Still needed to say if y zero is made it to emanated one. And you saw that this minute is going to be made it to again. So actually instead of all why not? That's for two less than negative one We have it. Why not these people to one bumpkin flying a little one one Why do you one is equal to native one and also it is. And that could be over. I have a limit. No one. So what if y zero is greater than one? See that wise here with one but one before That's why you one I don't see that y zero is greeted one well, less vise This, my dear less than two. Oh, you see that y zero to I knew it to and it wants integrated into do why finally like you rated likewise like that is equal to the case is not be exactly the same. Grab again. You have it. Why zero? Why zeros people too? Why is your He's really making too, and is less than one. It is no one. So, um, to like the less than one y zero people. One did you have? Why one we have that wise here was greater than one. Then you see that? So you know exactly sitting cases should be clear. Both of these figure out even. Certainly should I exactly with