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Problem 17 Easy Difficulty

Use a computer algebra system to draw a direction field for the differential equation $ y' = y^3 - 4y. $ Get a printout and sketch on it solutions that satisfy the initial condition $ y(0) = c $ for various of $ c. $ For what values of $ c $ does $ \lim $ $_{t \to\infty} y(t) $ exist? What are the possible values for this limit?

Answer

$L=\lim _{t \rightarrow \infty} y(t)$ exists for $-2 \leq c \leq 2$
$L=\pm 2$ for $0=\pm 2$ and $L=0$ for $-2<0<2$

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Video Transcript

things. So where supposed to use a computer algebra system to draw a direction? Field? Um, I like to use well from Alfa. It's a free online resource use for math help. So just go to wolfram alpha dot com and then in the circle bar here, you can go say direction. Field of Wipe Ron is equal to Why cute Minus for why? Like so. So press enter. Then we could get a direction field. There we go. Then you could just right click that Save the image in it out. Okay, so now that we have the direction field here, we need to put the, um, solutions for various y zero equal seat. So if you choose, like saying you know why zero equals zero by zero equals zero, which is, uh, sorry is right here. Then our solution curve will just look like a flat line. Here. We're safe. We go. Why is here equals one, Mamie, This will, uh So the lines are gonna go downward towards it like that if we start above too, you know, say three. Then our solutions will go up like that. Okay. If we start below zero. So here, Right are solutions will go, huh? Towards like that. And then if you start below negative too, our solutions will go down. Like so. Okay, if we start at exactly two. No. Sorry. A little lower than that should be down here. We'll actually get a flat line. Same thing. Negative too. We'll just get a flat line as well. Okay, So what values of sea does limit? Ah, why of tea exist? So why have tea last? He approaches infinity. Well, um, here you can see that we can get a value of but to hear zero or negative, too. So our values for this limit, uh, this limit actually exists when exes or C is between negative too. And positive, too. Okay, so we want see to be between, uh, negative too. And positive, too. Okay. And then the values for these limits are going to be here. This is zero to or negative too. So the values are negative to zero for two, and then we're done

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