 back at work. We are at lecture 19, entering chapter 7, Differential Equations. I did kind of make a count in here of the folks in this class. But if you are taking this class on cable TV this semester or in future semesters on DVD, there is a supplement that goes along with this book. And we're about to embark on the use of that supplement late in chapter 7, later in chapter 7. So if you don't have that, then I will, with your first test that goes back in the mail, I will include the supplement. And then there's also something, a summary of some chapter 8 things. I'll include those also. So if you don't have it, don't fret. I'll send it to you in the mail. Cable TV or DVD people. Chapter 7 deals with differential equations, probably in a way that you have never seen them, tried to visualize them. We have the derivative of something. We know what that is. By definition, we know it's a slope. And we're actually going to make pictures. And we're going to use pictures, thankfully, that are provided for us, that show what the slope is at individual points in the plane. So we know what the shape of the curve must look like, simply by the way the slopes go in the plane. It's called a slope field or a direction field. So we won't get into those today, but just to prepare you for the fact that differential equations have derivatives in them. That's why they're called differential equations. Normally, they'll have the function itself and a first derivative or a second derivative or third derivative or maybe some combination of all of those. So they have a derivative in them. And the order of the differential equation is determined by the order of the derivative that is contained in it. So if it has a second derivative in it, it's a second order differential equations. Not very many of those in the book. That's why we have this supplement to the book, because that does a more thorough treatment of second order differential equations. And to be very honest with you, the reason why we have this supplement is that the College of Engineering, which is a major client of ours. I mean, engineers are plenty in number and they populate our math classes, especially this calculus sequence pretty heavily. It's been determined that they need second order differential equations in this course before they take courses in engineering. So the book wasn't good enough, so we supplement the book. So primarily because of the College of Engineering. And it's good stuff. It's just normally not at this stage of the game. So chapter 7, section 1, differential equations. Keep in mind that there are entire courses on differential equations here at this university. Math 341 is intro to differential equations. It's a full three hour course taken after Calc 3. If needed for your particular curriculum, math 401 is the follow up to that course, which is an undergraduate level course, but it's part two of differential equations. So we're going to brush the surface of differential equations. And I know it's going to feel like more than brushing the surface, especially because of the supplement. But by no means is it a thorough treatment of differential equations. There are some techniques and types that we don't examine at all that many of you will see in math 341. So what is a differential equation? It's an equation that contains some unknown function and a certain variety of its derivatives. So it might be really, really simple. A simple differential equation might be that. We know how to handle that. Derivative of y is x cubed. We want to solve for y. We've already been dealing with things like that. We're anti-differentiating to get our way back to y. So technically, that is a differential equation. But we're going to examine some that are different than that. So these are kind of divided into categories here. This is an introductory section. We'll look at some solutions of differential equations. But we're not going to explore how we get to that solution. But we're going to validate that it is, in fact, a solution. So the first type would be population models. And starting with the differential equation, so let's say the rate of change of population is equals directly proportional to the population itself. So this is pretty prototypical first example of differential equation. It has the derivative of p in it. And it also has p in it. So the rate of change of population is directly proportional to the population. So what does that say about the population? Well, it's growing at a rate that the k value is going to be fixed. But as the population grows, doesn't the rate of change of population also grow? Does that seem logical for a population growth model? If there are more people there, we have more people that have the capacity to make more people to put it very bluntly, right? So as the population is larger, the rate of change of population is also larger. So the smaller the population, the smaller the population growth. If k is positive and the population is positive, it's kind of hard for the population to be negative, then we would expect the growth rate to be positive. So the picture that we would expect, and if this is with respect to t time, we would expect the picture to look something like this, that as the population gets larger, sorry, t is here, p of t is here, as the population gets larger, the rate of change gets larger, which means the slope of the tangent lines are steeper. It may not be quite this drastic. You might think, well, where's the rest of this picture? Well, it's over here, but we don't need that because this is t, which is going to be time. So it might look like that. It might be considerably flatter than that. It might just be slowly increasing. But as the population gets larger, the rate of increase gets larger. Another way of looking at this equation is we're talking about some population function, capital p, where if you take the derivative of p, it's a constant multiple of the original function, p. What is it that we already know about that looks something like this, such that when you take its derivative, you get some constant times the original function? What is basically its own derivative with the possibility of some extra co-efficient? e to the x. e to the x, right? So we would expect this to be exponential. This one with the dots over here and then going up here should look something like y equals e to the x or y equals 2 to the x. But we can always convert them to e to the x type functions. So let's say that capital p, this is what we're going to be able to solve by some techniques that we'll study in this chapter. But right now, we're kind of validating that this is, in fact, the solution. So we're allowing for the possibility of some arbitrary co-efficient. It might be 2 or 5 or negative 11. The k value is the same k that appears here. So if it is exponential, then it should, in fact, solve this differential equation. We can check it. Let's differentiate both sides with respect to t. So if we took the derivative of the left side with respect to t, we'd get p prime or derivative of p with respect to t. This coefficient, this number, comes along for the ride. What's the derivative of e to the kt with respect to t? k to the kt. e to the kt times k. So I'm going to take this k. It's all multiplication. So I'm going to bring this k and bring it out front. And then I'm going to have a c e to the kt. What is c e to the kt? p? That's capital P, right? So if we start with an equation, capital P equals c times e to the kt. And we take its derivative. Are we back to the differential equation that we started with? That is a legitimate solution. That's not how to get there. But at least it validates that that is a legitimate solution. So something of the form of c e to the kt is what we're talking about. We know that exponential functions, that if k is large, that it goes up fairly quickly. If k is small, it still increases. But the rate of increase is fairly slow. All right. So that is sometimes called uninhibited growth or uninhibited population growth. Because when you do sketch it, and it is this type of exponential function, it does keep increasing without bound. So it's uninhibited growth. Now not all things that grow grow in that fashion. For example, population. That may be a good model for the short term, but a not very good model for the long term. I don't know who's from the smallest town in this room. Who thinks that your town is the smallest? Jacob, what town? Matthews County, Virginia. Matthews, the whole county? Yeah. OK, so it must be small. If he's naming the county. All right, Matthews County. What's the town? I live in the town of Moon. Moon? Ooh, I like that. That's even better. So Moon, Virginia. Let's not do that, because that's against the law. The town of Moon, Virginia. So what do you think the population is? 114. I like Moon, Virginia. I'm going to go back and Google that and see where it is. I'm going to go there. Population 114. So if we use this model, so here we are, way down here, at time 0 today. The population is 114. Could we over the long haul predict the population? So here we are at time 0. Could we use this model to get the population of Moon, Virginia to be 500,000? What are the chances that Moon, Virginia is actually ever going to be 500,000? Probably isn't, right? There's a lot of things that will constrain that. The limits, I mean, the whole county, right, is only so big. The city limits are probably going to constrain that. Water, sewer, fire protection, police, all kind of things are going to constrain that. You'll probably never get that. But could we use this model to eventually get the population to be up here at 500,000? Yeah, you could. It's uninhibited growth. So when would that happen? I don't know. 2,500 years from now? If it could happen. So after a while, this is probably not a good model. So it's good for the short term. A couple years ago, I had somebody in class from Tarheel, North Carolina today. That's kind of small. I think it was not that small. Now the county's got like 9,000 of it, so. Oh my goodness. I'm just saying, I mean, I don't live in like Millinoware. Hey, let me tell you, middle of nowhere, I'm from a little town in Indiana, Speedway is the name of the town because the Indy 500 is there. And that's why the town got its name Speedway. If you haven't laughed at that yet, it wasn't 114, but it's a little bitty town in Indiana. Our high school, the nickname was the Speedway Spark Plucks. Because we were, yeah, that's pretty funny, isn't it? Legitimately, that was our high school nickname. So we were the plugs. The plugs. So I'm from a small town, so I know small towns. Now, so if that's a good short term model, what's a good longer term model, still population model. So let's start out the same way. And I'm going to put a letter down here that's in your book, and then I'm going to, because there are two of the same letter and the same equation, and I don't want that to be confusing. This is a lowercase k in the text, and they use a capital K here. And they're different things, so I'm going to switch this letter to L, a limiting value. They call capital K the carrying capacity, which doesn't start with K anyway, but at C normally means something else. So I'm going to switch this K to a capital L. So L is the carrying capacity. Nice to know how to spell that. Or the limiting value. So we've got this value that we're going to assign to Moon, Virginia, or Tar Heel, North Carolina, or Speedway, Indiana that there's only so much acreage there, so many square miles, water, sewer, all kind of constraining values that we're going to say the maximum population of Moon, Virginia is, I don't know, 3,000. The maximum population of Tar Heel, North Carolina is 7,000. Of Speedway, Indiana is 15,000. So we've figured that that's the maximum supportable population, so to speak. So we've established what that is. Now, what happens as the actual population gets very, very close to this limiting value? The change gets closer to 0. Change gets closer to 0, because if P and L are the same, or practically the same, then this fraction is practically 1. 1 minus 1 is 0. So that means the rate of change of population is pretty close to 0. So if P and L are the same, then the growth rate is 0. Well, the population then flattens out at that rate. So the picture, well, let me ask one more question before we get the picture. If the population is very small compared to the limiting value, so the population is 114, the limiting value is 100,000, this is not a very significant number. So it's 1 minus practically 0, which is 1. And then it just looks like, if this is 1, doesn't it look like the other model? So for small values of P, it looks like our old friend here, this exponential model. And as we get closer to this limiting value, it begins to flatten out. So this is a better long-term model for population growth. It's a logistic model. So it shows that initially, things are growing pretty much exponentially until we, at some point in time, there's a little point of inflection in here, right? Where it changes from being concave up to concave down. At some point in time, when we pass that, the growth rate slows down. It begins to flatten out. And as we approach the limiting value, the growth rate approaches 0. Now, there are two equilibrium values for this particular differential equation. What values for P would cause the growth rate to be 0? We already have one of them. When the population is capital L, then the rate of change of population is 0. So that means it's flat. So it's staying the same, equilibrium there. And when P itself is 0, if you look at the model, so these two are called equilibrium values. So if you start with P equals 0, you're not going to get this picture. If you start with P equals L, you're not going to get this picture. You're just going to get a flat line, right? Because the slope would be 0 at either of those. So there are some equilibrium values for this type of equation. We'll see them for some other differential equations as well. But this seems to be, this model, seems to be a model that's going to be good for the long term if we can somehow determine or make an intelligent guess about the limiting value. We will solve this eventually. It's in your book. A little bit later, we'll use a technique that will allow us to solve for P to see what this looks like. It kind of looks exponential in nature, but it's got a little extra baggage on the equation. But that's not part of what we're doing today. All right, another kind of differential equation that we'll use. And they're trying to get us to think about in this section what the solutions would be like. So the first time it seems that for population growth models, it seems like something exponential is going to work. And then we'll adapt that for the logistic population growth. Second thing they ask us to take a look at has to do not with the hooks law where we determine the force required to stretch or compress a spring, and then we figure out how much work is done. It's a different thing. It's the motion itself of the spring. So here would be the type of problem, the description of the type of problem. So we have this weight on a spring, and we pull the weight down and release it. And then the weight on the spring begins to oscillate. So it's the motion of the spring. Actually, it's the position of this object at the bottom of the spring that we're trying to track. So we've got this oscillation. We pull it down, we release it, it begins to go up and down, oscillate without any external factors like anything that's going to slow it down in any way. We would expect this spring to go from here, back up to here, back down to here, back up to here, and so on. Now, eventually, we would expect the amplitude of that to decrease if we throw in some realistic constraints. But if we don't, it would oscillate forever with that same amplitude. Now, if you take this over time, this motion of this object at the end of the spring, it starts here at equilibrium, it comes down, it comes back up, comes back down, goes back up, and so on. So we're letting the clock run here. We're letting T increase. So that is a better picture of the motion of the spring. I don't know. What's that look like to you? What do you think would describe this kind of signs? What else? Cosines, possibly, right? So signs and cosines seem to be what we're going to use to model something that's oscillating in this fashion. What's the differential equation going to look like? Based on two things, one of them is actually Hooke's law, but it's the restoring force, which is negative kx associated with this motion of this object at the end of the spring. The other is Newton's second law, which says that the force is equal to mass times acceleration. So if x of T is the position, then the second derivative of x with respect to T both times would be acceleration. So if we equate those, because they are both equations involving the force involved with the motion of the spring, we end up with mass times acceleration equals this restoring force and solving for the second derivative. So this is a differential equation. It's a second order differential equation, second order because it has a second derivative in it. So you could write this, I guess, x double prime equals some number times x. This, then, ought to give us some more ammunition besides the other picture where this thing's oscillating and the amplitude is the same and the period is the same throughout. What is an x thing? Excuse me. We've got an x double prime and an x in the equation. What can we take the second derivative of and get some negative coefficient times the thing we started with? Or let's look at it the other way. We're going to start with something. We're going to take its second derivative and it's kind of the negative of what we started with. Aren't those sines and cosines? If you start with a sine and you take the first derivative, first derivative is cosine, is that right? What's the second derivative? What's derivative of cosine? It's negative sine. So we started with sine, now we have negative sine. That's kind of what this equation is saying. Now, we're going to accumulate some other baggage along the way because it's not just going to be t. It's going to be 3t or 5t or some other number in terms of the t. So we'll accumulate some other stuff along the way. But the key thing is we started with a sine. We took the second derivative and we ended up with a negative of that sine. How about if we start with a cosine? What's the derivative? The derivative of cosine is negative sine. And what's the derivative of negative sine? Negative cosine. So we started with a cosine, we took the second derivative, and we have a cosine again, but we have a negative. If sines generate this and cosines generate this, how about sines plus cosines? And it does get worse, by the way. But if we take their sum, what's the first derivative? Aren't we just going to get the sum of these from right up here and right up here? Derivative of the sine, well, let's just track it down, is cosine. And then what? Negative sine. And derivative of cosine is negative sine. And then what? Negative cosine. So if we start with sines and cosines added to each other and we take the second derivative, don't we still get sines and cosines added to each other with a negative sine out in front? Is that correct? So that'll be very close to the model that we'll use in this situation, in this setting. Now, obviously, that's not reality. You don't expect it to oscillate exactly the same way forever, like sines and cosines would do. But we'll add those other constraints into the problem. General differential equations. So they might be as easy as, I wrote up here earlier, y prime equals x cubed. If that is what we're given for y, the derivative of y is x cubed, what is y itself? If its derivative is x cubed. What's the anti-derivative of x cubed? x to the fourth over 4. Wouldn't we include some more? We could include a plus c. So this tells us that we don't have a solution, but we have a whole bunch of solutions. Now, whatever we choose for c, when we work our way back to the derivative, it doesn't matter. If we choose over here 11 for c, is the derivative of this still x cubed? If we choose plus 8, the derivative is x cubed, no matter what we choose for c. So we've got this family of curves. And if we want to zero in on any one specific member of this family, we've got to have additional information. If this is all we have, we don't have an answer. We have a whole bunch of answers. What would be something that would give us that category of additional information? What do these look like? x to the fourth over 4 plus c. What if c is 0? What's x to the fourth over 4 look like? What's the slope? You graphed x to the fourth over 4. x to the fourth looks like a kind of parabolic shape, but it's skinnier. Things go up pretty quickly. So let's say x to the fourth over 4 plus 0 looks like that. What if it's plus 1? What if it's plus 8? And so on. So what do all these curves have in common? Well, they all have the same derivative. The derivative, at any point, is the cube of the x value. How about at x equals 0? Slope is 0, slope is 0, slope is 0. So in that sense, they're kind of parallel curves. How about at x equals 1? What's the slope? 1, right? It's the cube of the x value 1. So we come over here to where x is 1. The slope here ought to be 1. The slope here ought to be 1. The slope here ought to be 1. So again, we've got kind of parallel curves. It is true that for all these curves, regardless of what we choose for c, the slope of the tangent is the same. So the constant doesn't matter. We know that derivative, the constant of 0, disappears. So back to my question. What other type of information would allow us to zero in on one specific member of this family of curves? What if we said it contains a certain point? That when x is 1, y is 11, is that one of these curves? When x is 1, y is equal to 11. We would take that information, plug it into this equation, and solve for c. Find the 1 value for c such that when x is 1, y turns out to be 11. I just made up a point. So let's see how much sense it makes. Probably not a lot. So we've identified a point that this curve, one curve, one member of this family has to go through. So y is 11. When x is 1, so the only variable is c, we'll find the one member that goes through this particular point. So what is c? 10 and 3 fourths? So this is no longer a family of curves, but it is one member of that family of curves that goes through that one specific point. That point being 111. A lot of times you are given an initial value. So you'll have a function of t. And it will say when t is 0, what is the population at time 0? What is the number of bacteria in this culture at time equals 0? So you'll be given some initial value. So if you have t equals 0, population is 120,000. So there's an initial value that's going to take this family of curves and get it down to where it's one single member of that family of curves. So that's pretty common. In fact, let's get an example all the way through. I think we have just the right amount of time here. How many times last semester did I write on the little taped thing? Not going to do that this time. I might be from Speedway, Indiana, but I can learn. Got to watch my alma mater on ESPNU last night. Indiana State was on ESPNU where I went to college. Just a couple of years ago. No, a long time ago. What's Indiana State's nickname? Who knows that? Indiana's Hoosiers. Indiana's? Indiana State. See, that's how popular Indiana State is. The Sycamores. Who's the most famous other than the one you're looking at right now? Who's the most famous Indiana State alum in athletics? Beats me. Larry Bird. Really? You heard of Larry Bird. Of course, you guys, I mean, Larry Bird was winning. You guys were born and win. 90? 90. Yes, of you. Ash Bird. Ash Bird won the NBA in 1986 and 1987. You guys weren't even born yet. Did you play basketball there? I did play basketball one year and played football four years. But that my claim to fame in basketball is that I have played numerous pickup games with Larry Bird. Nice. That was fun. That's crazy. Not that I deserved to be on the court with Larry Bird. I ended up in the gym and they needed somebody else. And I was fairly athletic at the time. So I got to play pickup ball with Larry Bird. That was fun. OK, here is a differential equation. So it has y in it. Y is kind of the unknown function. It has the derivative of y in it. So it's a first order differential equation. Here is the solution. So again, we're in that stage where we are going to be handed the solution. Notice there is something in the solution that makes it a family of curves as opposed to just a single curve. Now, as we do different techniques, you're going to see the C and the K values appear in different places. Used to be we would integrate and we'd just put a plus C at the end. And we'd always be OK because that's the constant, the derivative of the constant with 0. Now notice that the plus C is in the numerator and both of these terms get divided by x. So you'll see the C's and the K's in different positions. So is this a solution to this differential equation? Let's verify. How do we verify? Which is part of, I'm sure, that's what the web assigned questions are because that's what the questions are at the end of this section. Verify or validate solutions. Plug it in for y. Plug it in for y and do what else? Solve it for y. We need to find its derivative and throw it in there and then put the original function in there. And if it is an equation, then it must be a solution to that differential equation. So we have y. Let's find y prime. y itself is a quotient, so it seems appropriate to use the quotient rule, denominator, times derivative of numerator. What's derivative of numerator? 1 over x. 1 over x derivative of the c is 0 minus numerator times derivative of denominator all over denominator squared. So let's see if we can simplify. x times 1 over x is 1. Distribute the 1 as if that's going to take a lot of work. So the minus 1 here and the minus 1 here. So there is y prime. So let's plug these things in. Plug y in here and y prime in here and see if that is, in fact, a legitimate equation. So x squared times y prime, that works out nicely. Plus x times y is that equal to 1. So we're kind of verifying the solution that's been given to us. How's it look? Yeah. I think it's going to work. The x squareds knock out. The x's knock out. We've got 1 minus natural log minus c. Add back in the natural log. Add back in the c. So left side and right side are, in fact, the same thing. So we have validated or verified a solution. Here's the solution to the differential equation that was given in RET. Now you have another good example in the book on page 502 that is kind of along the same lines as far as this level of difficulty. So let's backtrack to that solution. Let's suppose we don't want a family of curves, but we want a curve. We'd have to have some additional information. So let's say, and I'll use the author's notation here, that the y of 1 is 2. So when you put x equals 1 into this function, you're supposed to get y equals 2. So the y of x equals the value here. So when x is 1, y is 2, what was our function? Our solution to the differential equation was that. We don't want a family of curves. We want a curve. So call this, if you want to, an initial value. It's a value that we can solve for. So y is 2 when x is 1. Plug both those in, keep the c in the equation, and solve for what c is. Natural log of 1 is 0. So it looks like c is 2. So the one member of this family that we want to address with this particular constraint is this one. No longer a family of curves, but a single curve. Let me reinforce something that I know. I just know it gets lost sometimes in the middle of the semester because you are busy people. But while I was given a test to the distance ed people that are taking this class on cable TV, I didn't take enough work with me, so I had the book. So I reread the book through a lot of chapter 7. This is pretty good. I mean, it's not leisure reading. I know that. It's not like Nicholas Sparks and all these romance things that you're going to read. But it's pretty good. Pretty easy reading. So I guess don't let me or WebAssign or what happens in here be the only thing that you get out of the book. It's very well written. It doesn't take that long. I'm trying to do examples other than what's in the book, so I think it's to your advantage to also look at the examples that are in the book. So when you have time or try to make time, especially if something's completely new for you, which it might be in this chapter, make sure that you read the book. It's well written. You paid a lot of money for it. Let's get some use out of it. We are done for today, and I will see you tomorrow.