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. Thank you and have a wonderful day.