Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. A solution curve of the differential equation Y prime equals some function in terms of both X and YF of X. Y is a curve in the XY plane whose tangent line at each point XY has slope M equal F of the function X Y. Slope fields. It's a visual suggestion of the general shapes of solution curves for a differential equation. So a slope field. What we really do is we look at each and every point XY and we put in what the slope is at that specific location. We usually use a symbol of just a very small line segment, so once we get this whole picture of what's happening at each of these points, then we can get a general idea of the shape for the solution curves. A general equation for an object thrown straight down is the acceleration equal DVDT, which is equal to G minus KV, where A is the total acceleration, G is gravitational acceleration, and K is a constant force. Frequently K is thought of as .16. K is the air resistance proportionality logistic differential equation DPDT equal KP times the quantity m -, P If we just distribute that out, we get KMP minus KP squared, where P is the population that inhabits the environment. Sometimes it's thought about as the initial population and M is the maximum population the environment can sustain. It's actually going to be the asymptote that we approach when we get really, really large for our value. So then KM is going to represent the decimal of the growth rate. So if it's .06, it's a 6% growth rate and negative KP squared represents the inhibition of growth due to limited resources in the environment. What a logistic differential equation can do is say that we start with a lot of whatever, a lot of mosquitoes, but the environment doesn't have enough people to feed all these mosquitoes. So if we start with too many, what will happen is we will decrease to some substantial quantity that this, the population, the environment can sustain. If we start with too few, then what's going to happen is those are going to grow and grow and grow until we hit that, whatever that number is of the maximum we can sustain. Theorem 1. Existence and uniqueness of solutions. Suppose that both the function F of X, Y. This is really just a function that's in terms of both X&Y. So it's no longer F of X equal, it's F of X&Y and it's partial derivative dyf of XY are continuous on some rectangle R in the XY plane that contains the point AB and its interior. Then for some open interval I containing the point A, the initial value problem DYDX equal F of X YYA and YA equal B has one and only one solution that's defined on the interval I. Now it's important that we understand that this is only saying that there's a unique solution at this location or very close to this location. It does not say it's a unique solution all the way throughout. The other thing we want to pay attention to is if continuity fails, then the initial value problem may not may have no solutions or infinitely many solutions. So on this first example, they're wanting us to draw in a slope field. So we're going to start with the given point and we're going to go with the direction of all of these pieces that are coming together. And as we do these pieces, we're going to flow and the direction that these pieces are going until we connect with the curve. And we're going to do that for each of the dots. So we're going to flow with the slope field. This next one here kind of between here, we're going to flow with the slope field. We're going to do this with all of these and it gives us a concept of the direction based on the given information as to how we would go about solving for the actual equation. I got a little off there actually going to go. I think you can see what's going on here and complete the rest. Our next example, we're going to be given a function and an initial condition and we're wanting to know whether existence occurs at the solution and whether it's guaranteed and if so, if it's unique. So we're going to start by saying that the given equation is really going to be our F of X equation in terms of F and fxy. So if we let fxy equal to X ^2 y ^2, then we're going to take the partial in terms of the Y. When we take the partial in terms of Y, we're going to get four X ^2 y and we can see four X ^2 y at the initial condition would indeed exist. So we have its continuous everywhere. It's never undefined in this equation and thus uniqueness occurs. So this one was a pretty simplistic one. We set the original derivative as the function in terms of X&Y and then we just take the partial in terms of Y. The fact that this exists and is continuous everywhere tells us that uniqueness does occur. If we look at this next example, we're going to start once again by saying our, our function, our condition in terms of X&Y is the cube root of Y. So that now when we take our partial, we can see partial in terms of Y, we can see that we're getting 1/3 Y to the 2/3. So this FXY is continuous. So existence occurs, but the partial is not continuous. And where is it not continuous? It's not continuous at this condition of our Y 0 being 0. So it's not continuous, thus not necessarily unique. Might be new unique, but it's not necessarily unique. So let's look at one more of this type. If I have dy DX equaling sqrt X -, y at Y-22, then we're going to start by letting our function in terms of XY be that derivative. And then we're going to do the partial in terms of Y, and we're going to get -1 / 2 square roots of X -, y. Now, if we start, 22 is an end point because we know that inside that square root over the real numbers has to be greater than or equal to 0. So X would have to be greater than or equal to Y. But that's going to occur with our 22 conditions. So that's where sqrt 0 is. So it's an end point. Thus, it's not continuous because it's got a starting point, doesn't go on and on in both directions, and the partial is not defined, so no guarantee on existence or uniqueness. Now the next one we're going to do, they're giving us Y prime equaling X plus YY 0 equaling 0, and they want us to find Y -4 equaling what? So what we're going to do here is we're going to make a pretty detailed grid so that we can actually put slopes in at each of these various locations. So if we know that the slope or our Y prime equals X + y, we're going to have X + y. So when we're given X + y, we're going to get -10 negative nine -8, etcetera. The -5 + 5 would give us a slope of 0. So we know that at -5 positive 5 we're going to have a horizontal line, and at -5 negative 5 we're going to have a pretty steep slope of 10. So something like this, we're actually going to fill in this entire grid. We know that at Y0 we have equals 0, so we know that that's actually a given point on the actual graph and we're wanting to find Y of -4. So the first we're going to do is fill in this entire grid and then we're going to draw in our slope fields with these grid points and it has to go through that 00. So I actually have done this already. So I filled in the entire grid. I put in little tick marks based on the slope and the steepness, and then I knew it had to go through the .00. So as I'm drawing in my graph, I have to go along the slope fields, but I had to go through this point. So if I'm here, I'm going to come back up this way based on the slope field. So now when we look at Y of negative 41234123, about 3 1/2 maybe that's obviously an approximate based on my graph. Thank you and have a wonderful day.