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Problem 12 Medium Difficulty

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.
$ y' = 1 - xy, (0,0) $

Answer

Sketch the direction field of the differential equation $y^{\prime}=x y-x^{2}$
And we have to sketch the solution curve passing through $(0,1)$
Direction field and solution curves are shown below:

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Video Transcript

you're getting differential equation and the point plane. Yeah, find And the solution at this point, find this look. First participation is white crying because one minus X one. So for a given value of white crime, white pine is equal to see some seen real numbers one night. Why? So that that's why people to one night I see or why is equal to We'll see that this is the reciprocal function one of her, but by a factor of won nine seats. Grab this. Look. Field y axes on the scales tonight. More information from represent. You see that? A long term Klein was one of Rex. You have C zero. Most of these changes are one to the euro looking flat. So particular at the point 11 you have want 1/2 to another flight. And so between all these on his hurt, you will have flat second, also ahead. That may be of 11 Well, as negative half native to and so one. Now it's easy. Positive one. So that I spoke with him one. Well, this only along the line light with Europe. So you accept really have every point had it one move it hard represent graph. Never seen him on the X axis. No slip of Warren is going to look. Si is equal to negative one. This corresponds to all the points on occurred while wolves two over which is her white with one of Rex by a factor of two. So thank you. 0.12 I have a slope. Negative one Looks like usual again. Likewise. But one 0.5 four one negative one. So one likewise the point. Negative one Number two a so called negative one point made a point by no, no name one. You can see that as his hair purple is get farther and farther away The organ on the right is going to get more and more negative. Okay, this is the approximation. No defects that you What sort of business? A drag going on Like Waas. Get further with your partner. So getting more and more So you have a rough idea of what it looks like are the points quadrants? 13 Now some points that were missing. Quote like point by one. If we have a 51 are getting too equation y crying too. One minus 5.5. I don't consider that slope before 0.5 except certainly people way. So we'll see. We have more. Take five. This is his wife. This is never home Kribel of But you press. Is it yours again? This is a but will be. You have to be exact. We're important too. That may happen. X is less the zero. Why? Great deal? A negative x Y zero. So we suppose that X is equal. For example, by one along the Berg lack We have one kind of like along the exact Get a Super one. Now we have I want to Who? One of your function across the x axis of light. This looks wanted you particular one. Maybe one? No. Why? What by so long? So we see that as we get herder and further away from Oregon, these organs slow is going to get rep approximation. They get the slope, gets steeper and steeper as we go on. Privileged figure. Do you have any sections in between? No, this is a tubes between 21 My wife. For whom? One. This is a pretty good picture of our field. We have sort of this kind of diamonds form here, around origin. What have you? Mm. But you see me, we have sort of diamonds pattern here with all the folks that are close to zero contained in this dining origin in order to find the current zero first. No point years here, Kurt about agree, get to increase of orbit and suddenly increasing again to it's sort of follow a lot. Whatever. You know, all the payments on that section has this look of zero. Likewise, as the you see that graph creasing in the Y direction a little bit, and then that's too increase until the line paper below one of her ex again. We have this sort of Sinus soil shape my side.