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Current time:0:00Total duration:7:41

Exponential models & differential equations (Part 1)

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Video transcript

what I'd like to do in this video is start exploring how we can model things with the differential equations and in this video in particular we will explore modeling population modeling population we're actually going to go into some depth on this eventually but here we're going to start with simpler models and we'll see we will stumble on using the logic of differential equations things that you might have seen in your algebra or your precalculus class so on some level what we're going to do here is going to be review but we're going to get there using the power of modeling with differential equations so let's just define some variables let's say that P is equal to our population and let's say that T is let's say that T is equal to the time that has passed in days in days it could have been years or months but let's say we're doing the population of insects that reproduce quite quickly so days seem like a nice time spits banned to care about now what would be a reasonable model well we could say that the rate of change the rate of change of our population with respect to time with respect to time is well a reasonable thing to say is that it's going to be proportional to the actual population the actual population why is that reasonable well the larger the population the larger the rate at any given time if you have a thousand people the rate at which they're reproducing is going to be more or a thousand insects it's going to be more insects per second or per day or per year than if you only have ten insects so it makes sense that the rate of growth of your population with respect to time is going to be proportional to your population and so you know sometimes you think of differential equations is these daunting complex things but notice we've just been able to express a a reasonably not so complicated idea the rate of change of population is going to be proportional to the population and now once we've expressed that we can actually try to solve this differential equation find a general solution and then we could try to put some initial conditions on there or some some states of the population that we know to actually solve for the constants to find a particular solution so how do we do that and I encourage you to pause the video at any time and see if you can solve this differential equation so assuming you at least maybe have added attempt at it and you might immediately recognize that this is a separable differential equation and in separable differential equations we want one variable and all the differentials involving that variable on one side and the other variable and all the differentials involving the other variable on the other side and then we can integrate both sides and once again DP where the rate of the derivative of P with respect to T this isn't quite a fraction this is the limit as our change in P over change in time this is our instantaneous change but for the sake of separable differential equations or differential equations in general you can treat you can treat these this derivative and liveness notations like fractions and you can treat these differentials like quantities because we will eventually integrate them so let's do that so we want to put all the peas and DP s on one side and all the all the things that involve T or that I guess I just don't involve P on the other side so we could divide both sides by P we could divide both sides by P and so we'll have 1 over P you have 1 over P here and then those will cancel and then you can multiply both sides times DT we could multiply both sides times DT once again treating the differential like a quantity which isn't it really isn't a quantity you really have to debut this as the limit of as that change in P over change in time the limit as we get smaller and smaller and smaller changes in time but for once again for the sake of this we can do this and when we do that we would be left with 1 over P D P is equal to is equal to K DT is equal to K DT now we can integrate integrate both sides we've because this is a this was a separable differential equations we were able to completely separate the P's and DPS from things involving T's or I guess the things that aren't involving P's and then if we integrate this side we would get the natural log the natural log of the absolute value of our population and we could say plus some constant if we want but we're going to get a constant on this side as well so we could just say that's going to be equal to that's going to be equal to K it's going to be equal to K times T K times T plus some constant plus some constant I'll just call that c1 and once again I could have put a plus c2 here but I could then subtract the constant from both sides and I would just get the constant on the right-hand side now how can I solve for P well the natural log of the absolute value of P is equal to this thing right over here that means that's the same thing that means that the absolute value of P that means that the absolute value of P is equal to e to all of this business e to the e to the let me do the same colors K T KT plus plus c1 plus c1 now this right over here is the same thing just using our exponent properties this is the same thing as e to the KT AE to the K times T times e times e to the c1 e to the c1 now this is just e to some constant so we could just call this let's just call that the constant C so this is all simplified to C e C e to the KT to the K T and if we assume our population at any given time is is positive that we could get rid of this absolute value sign and we have a general solution to this frankly fairly general differential equation we just said proportional we haven't give what the proportionality constant is but we could say if we assume positive population that the population is going to be equal to some constant C times e to the KT power to the KT power and the reason why I said that you've seen this before is this is just an exponential function and it's very likely that an algebra or in precalculus class you have modeled things with exponential functions and my guess is that you've modeled things with populate model things like population the reason why this is interesting as you now see where this is coming from the underlying logic that's just driven by the actual differential equation the rate of change with respect to time of the population well maybe it's just proportional to population so I'll leave you there and the next video well we'll do what you probably did in the 10th or 11th grade or maybe later in your life it doesn't matter when you did it where we actually look at some conditions to find a particular solution