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Sketch a direction field for the differential equation. Then use it to sketch three solution curves.

$$

y^{\prime}=x-y+1

$$

Draw the solution curves such that the slopes of these lines will be tangent to the curve.

Differential Equations

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Campbell University

University of Michigan - Ann Arbor

Boston College

So equation find three differential equation Is white crime before Why? Plus one. Now we see that we have thanks your numbers that why prime or seen? It's why. Why plus one or one people to what one minus? This is the equation of a line one. Why set 19 c Grafting Sofield in particular. It takes me one every a slump of one, um, line white. All of you slopes on that line one. So over the line like that. Then we'll get slip 100 of one again at 11 Want again at 22 points on that line and so on. We have that seeing same quarter zero. Then we have the lying like those experts. One all the points that are going to have slowed to zero. But with 001 You 23 he said. One fill in one. You It will take scene to se When you get the line wine. It's once one for hard line is that maybe one this will be slow to si is equal to three. He sees a negative one se we'll get the line. Why? But to no down zero to negative one this line. Maybe two. It is. We get a line by equals X plus three. A very steep line. Zero. Really? You know, along that line? Yeah. He's downward slopes. I think you see a pattern here so that as we get further and further away from the line, why was X plus one? Our slopes? It's steeper. You're almost vertical. This is a very rough throwing what slope looks like. Fortunately, my laptop is cooperating right now a little bit years and be a very steep, credible one. You, almost a little. This is a very rough sketch. So you man testifying Three solutions on a slow field, for example, pollution had initial value 00 graph. It's in black. No, this is a point at 00 in If we follow the slopes of this serious possible, see that they're flat now. A little bit as we approach expertise. Infinity on the right. That's perfect. Get this heard. It's very speak, really going down. It looks a little bit like involved with perhaps shipment. However, it doesn't have any birth blasting money. Grab draw had initial solution y zero people's one. Grab this bread euro one Follow possible experts. Name goes off and that's the coaches. Starts out level and words they were to our other breath. Finally, the explosion will have value y zero hold a negative one. So we know gratis every one Happy euro. Negative one on you See that it's very quickly. Well, and that's purchased positivity this slower, more slowly until it's running about the same rate as the other two grand.