This is why you're learning differential equations

this video is sponsored by brilliant let's just get to the point why do we learn differential equations because they are the equations that describe how nature works the universe one day said boom here's a bunch of stuff and we said cool how does it all work and the universe said differential equations they're tough often impossible to solve exactly but you'll eventually get some cool stuff from them modeling how a population will grow involves differential equations how any fluid moves differential equations electromagnetism which is how phones radio Wi-Fi and GPS all work not one but four differential equations known as Maxwell's equations there's electric circuits and even if something touches something else and thus exerts a force differential equations are used to describe the motion if there's no physical contact like with orbiting bodies there's still a force so there's still differential equations now as with simple algebraic equations differential equations can and often do have meaning when you read into the story that they're telling like this equation could be saying my age is y and I'm exactly five years older than my brother whose age is X and we can use this to determine either of our ages given the other pretty boring but we can give it meaning this equation can have meaning too or rather it asked the question and that question is what curve or family of curves have the property where the numerical value of the area under the curve is twice numerical value of the arc length on that same interval for any given interval A to B well to set this up we start with this equation which is the area under the curve itself and this is the equation for the arc length on that same interval A to B but we want to know when the left side always equals twice the right side now we can differentiate both sides to remove the integral sign and we're left with a differential equation from here we can square both sides and then distribute and we get that original equation if you solve for that function y you get this which is the curve you're seeing here so that's one example of meaning within a differential equation but let's see how these really describe some real-world situations because it's not always obvious what story these equations tell or how they show up in general and since I always love bringing up the tv-show numbers when I can here's a perfect example so check out this scene from an episode where law enforcement is trying to catch a couple committing crimes as they travel across the United States we plot your movements against those targets the pattern makes itself known and when we plot your path against oil and winters we got this yeah this is the Red Desert Robbery the missing point on a curve that I didn't even realize I was looking at it's a variation on something called a pursuit curve alright so what's happening here is the FBI has been chasing this couple across the country but hasn't caught them yet so the mathematicians plotted the path both the criminals and the FBI took to see if they can make some predictions about where the criminals will strike next and this is related to pursuit curves a pursuit curve is simply the curve traced out by one object chasing another although there are usually conditions listed for something to be technically considered a pursuit curve but this could apply to a cheetah chasing down a gazelle or one aircraft chasing another for example so let's see how we can determine the curve that the pursuing object will trace act now two things to note first we're going to assume that the plane being chased has a predetermined path whether it's flying straight up or in a circle or whatever assume we already know their path then the second assumption like I mentioned before is that the chasing object is always moving in the direction of the other and it turns its nose as needed during pursuit in reality this could be like a plane wanting it's forward-facing guns always aimed at the target or something anyways what you're seeing here is a snapshot of the chase and we can represent the current positions of each plane with a vector I'll say C and M for cat and mouse the other thing we know is that at this time or really any time the chasing plane is pointed at the front one which means this is their velocity vector or C prime at this moment there's no wind or anything so the plane is definitely flying in the direction it's pointed now we don't know the speed or the length of that vector currently but we know the direction at least that's more important because there's another vector pointing in that same direction which is easy to find and that's M minus C for the visualization M minus C is the same as M plus negative C and putting negative C on top of M gives us a vector that points from the back plane to the front one then to deal with the lengths I'm gonna normalize the two vectors by dividing by their magnitudes so they both have a length of one this is key because the dot product of two vectors both with length one and pointing in the same direction is one this is a fundamental equation yes I could have just set the vectors equal to one another but I'm doing this because in that episode when the mathematician is explaining pursuit curves you can see that equation come on the screen so now you know the meaning behind it but still it's not really obvious how to solve it yet but if we go back to our snapshot remember that the N vector is actually a known function of time we're seeing it only at one moment in time but it is changing as the planes fly around so I'll say it's a vector function U of T comma V of T which are both known I'm just keeping things generic the C vector also changes in time but it's our unknown some X of T comma Y of T that we want to solve for and the C prime vector would just be X prime of T comma Y prime of T so then M minus C would give us this vector here just the X components subtracted and the Y components subtracted and I'm just not writing the t's so there's room I'll do the same thing with C prime then if we plug all of those into our equation that we want to solve we get this now the one extra thing I did was set the magnitude of C prime to 1 which just means we assumed the chasing plane is flying at a constant speed of 1 just to simplify the equation then we just have to do the dot product which leaves us with this differential equation you'll notice I actually wrote out the expression for the magnitude of M minus C on the bottom here the only problem with this is that there are two unknown functions x and y which means we need another equation but that would just be the one saying the chasing plane is flying at a constant speed of 1 now we have a system of differential equations that can be solved if for example we assumed the target plane is flying straight along the y axis then this would be the path of the chasing plane shown in red if the target plane we're flying in a circle then you get something very different shown here on brilliant sight now you'll notice this method of chasing someone isn't necessarily ideal for catching them but there seems to be other types or variations of pursuit curves that range from always aiming at the target to predicting where they'll go next on that numbers episode the situation was more complex as the mathematicians were accounting for how the Meuse of the FBI might affect the criminals that were being chased but still the basics of pursuit curves can be seen in a first level differential equations course and while this might not be a real-world FBI case pursuit curves can be applied to missile guidance systems aircraft submarines and so on all right now let's look at something more casual if you go to the gym you may have seen or done a bench press or squat with chains hanging off the sides this makes it so that as the bar moves up the chain is more and more suspended and this contributes more of its weight to the exercise meaning as the person pushes upwards the weight increases this actually complicates the equation of motion more than you might think because for every little DX or change in height yes I'll be using X as the variable there's a DM or small change in mass in regards to the part of the chain that's off the ground so let's see what this set up would look like now we'll say the barbell has no mass just to make things easier so really we're just lifting the chain off the ground and we'll call that distance off the ground X measured in meters then let's give the chain a way to density of 10 Newton's per meter thus the weight of just the section off the ground is 10 X so if the chain were 2 meters off the ground then you'd have to use a force of 20 Newtons to hold it in place then the equation for mass for the part of the chain off the ground is simply the weight divided by gravity weight is 10x and I'll round gravity to 10 as always so the mass of this part of the chain off the ground is just X lastly we'll say the person is pushing up with a constant force of 50 Newtons meaning the net force is 50 up minus the 10x down from the chain itself okay now before we can move on we have to realize that F equals MA is a lie well not really but it only applies to special cases where the mass is constant the real equation we have to work with is f equals the rate of change of momentum or mass times velocity doing a simple product rule we get this here and you'll notice when M is not changing which we get used to in a first level physics course then this term is zero and we're left with F equals m times dv/dt or F equals MA as usual but with the changing mass we have this entire equation from here I'm just going to replace the variables like mass is really X and force is 50 minus 10x so we get this here but velocity is just the rate of change of position and DV DT which is acceleration is the second derivative of position so we're left with this equation here moving the 10x over we're left with a second order nonlinear differential equation solving this would not be easy but not really the point of this video instead I just want to highlight that the motion which results by something as simple pulling a chain upwards has to be expressed through a not so simple second order differential equation and while it may be true that analyzing the motion of a barbell isn't too applicable the idea of analyzing a system with changing mass is the perfect example is a rocket as exhaust leaves the bottom the rocket itself loses mass little by little so thrust and a changing mass are kind of linked and the situation also leads to differential equations but now let's look at the most real-world application I can think of this here is a curve of the number of currently infected people from the corona virus as of early June it's infected people versus time which means the instantaneous rate of change or slope at any point is di DT it gradually increased for a while before flattening out but we want that rate of change to go negative anyways this is an incomplete picture of what's going on because there are other categories of people out there in the population there those that have never been infected or those that are susceptible those that are currently infected and then those that are recovered or unfortunately deceased but the most basic model the SI R model just calls it recovered and assumes no deaths I know many youtubers have talked about this recently so I'll keep it kind of short in the case of the corona virus everyone started in the susceptible bucket and let's say the population is 20 so no one is infected but everyone has the potential to be and then one day one person transition to infected this was basically an initial condition now if the population stays constant no one new is born then the number of susceptible people can only go down or stay the same because as soon as you get infected you leave that group never to return assuming immunity after you get the disease so the rate of change of susceptible people is going to be negative something it can only go down since having a lot of infected people or a lot of susceptible people / a high population increases the magnitude of that change then we say it's s times I where s and I are the number of susceptible and infected people respectively if either of these are large and the other is nonzero you have a large transition to those who are infected but we also have to include a constant that constant depends on the virus and us as it scales how quickly people go from healthy to sick social distancing or hand-washing for example would decrease that constant and this rate of change wouldn't be as extreme then the rate of change of those who are infected starts with that same expression seen on the left but positive this is because in this model it's a one-way street if someone leaves the susceptible pile they go to infect it but people will leave the infected pile proportional to how many infected people there are this constant in reality is death rate combined with recovery rate if medicine is released that speeds up recovery by 50% and that constant goes up and people are quicker to leave the infected pile and become recovered the rate of change of those who have recovered can only be positive or zero and it's that same constant times I you'll notice in this model that we kept the population constant so the sum of the three categories was always the same meaning the rates of change should add to zero as they do but here we're left with another system of differential equations that when solved will tell you how an infection will spread through a population in this simplified scenario I'm sure many of you have seen number files video on this that I'll link below but there you can see what happens when you play around with the equations and constants and all that as with most of these examples we did simplify things to make the math easier but this is all the foundations of what's going on in the real world if you go to the website for the International Council for industrial and applied mathematics there's a page on the mathematics of koban 19 showing the simulations and models that mathematicians are creating to understand the spread of this virus in different parts of the world where you'll find equation just like we saw equations very similar to these also show up in terms of population growth one of the first differential equations a lot of us learn is the one that models how a population will grow at a rate proportional to its current value the more people or animals there are the faster the growth as expected but things get more complicated with deaths or when maybe one population kills off another here's an example of bacteria which can multiply on their own and phages that essentially feed off bacteria and thus will die without them and you'll see that after one cycle here the phages grow from 1 to 4 but the bacteria stay at 2 and now there are more phages so some will die while others will continue to multiply so there's kind of this back-and-forth that happens but it leads to differential equations where the rates have changed depend on multiples of the populations and it's not always about just solving the equations as there are tools such as phase portraits that can help paint a picture about what's going on without having to find an actual solution here if you're given a certain population of phages and bacteria this would tell you how the system or really both populations will change at that moment and with this we can find equilibrium points or long term behavior for example and while I'm not gonna go any further than this think we've discussed a lot if you want to dive more into the topic of differential equations you can of course do so right here at brilliant currently they have two differential equations courses which are some my favorites because of how much they focus on real-world applications they have the pursuit curves we discussed there's 2d and 3d wave equations there's the equations that model the behavior of beams and much more their first course does start at the basics for anyone just starting out but by the second course there are things I never saw as an engineer in college so regardless of where you are in your education there likely is a lot to learn whether you want to get ahead as a student or just brush up on old topics so if you want to get started right now support the channel you can click the link below or go to brilliant org slash Zack star plus the first 200 people to sign up will get 20% off their annual premium subscription and with that I'm going to end that video there thanks as always my supporters on patreon social media links to follow me are down below and I'll see you guys in the next video

Channel: Zach Star

Views: 1,261,216

Rating: 4.9511304 out of 5

Keywords: zach star, zachstar, differential equations, the applications of differential equations, why you're learning about differential equations, pursuit curves, numb3rs, mathematics, applied mathematics, applied math, math, population growth

Id: ifbaAqfqpc4

Length: 18min 36sec (1116 seconds)

Published: Tue Jun 09 2020