 Hi, I'm Zor. Welcome to a new Zor education. I would like to extend our ordinary differential equations theme towards higher-order equations. Higher order in terms of derivatives which participate in the differential equations will be of higher order, like second-order derivative, third-order order derivative, etc. Well, actually I'm not going to talk about third-order or fourth-order. Probably the second-order is something which I would like to restrict myself. And the reason why it's important is that the second derivative is extremely important for physics, mechanics primarily. And I'm just going to demonstrate what's the relationship between the physics and the second-order ordinary derivatives. And this relationship is expressed in the term of acceleration. Acceleration. Now, when I was talking about derivatives, I basically related the first derivative as a rate of change of, let's say, distance from some object or from some beginning of the motion as the object is moving forward. So the distance is increasing, let's say it's increasing, and the rate of this increase is basically the speed at any moment. So if you express your position, let's say we're talking about a horizontal movement on a coordinate axis and this is position at moment t. Now, the distance from some kind of origin of coordinate is distance x of t. Then the first derivative at this particular moment is the speed. This object is moving at this particular moment because it can change. At the next moment, it can be different. Now, what is the acceleration? Well, that's the second derivative. This is a rate of change of the speed. So that's basically a definition. So we have a first derivative from the function which represents the distance as a speed and the second derivative, which is derivative of the first derivative. So this is x of t, second derivative, is acceleration. Okay, fine. So we basically have defined the acceleration as the second derivative of the function which characterizes the position of our point on the x-axis. Okay, that's fine. Now, let's go into physics, into mechanics. And I would like actually to talk about the second law of Newton. So I hope you know that there are three fundamental laws suggested by Newton in very, very long time ago, like 17th century or something like this, which basically put the foundation of the whole classical mechanics. And the second law is the following. Force is related to mass and acceleration by this formula. Well, formula is fine, but let's just analyze this formula. When it was introduced at some moment, people were primarily thinking about, okay, let's apply a constant force onto a constant mass. And then it would cause certain acceleration of the movement of that mass, which is constant A related to force and the mass in this formula. And this is fine. What I would like to introduce is, let's just go a little bit further. Let's interpret this formula not as the case where constant force, constant mass and constant acceleration are related to each other, but variable. Because the force can be a function of time. The mass can be function of time and acceleration can be function of time. Well, let's forget for a second about mass being variable. Let's consider there is a constant mass. Let's say you're driving a car. Now, when we're talking about force, we're talking about the force, which is actually not just one single force, but a balance of all the forces which are affecting our object. Now, what are two major forces? One is the force which our engine is actually pushing the car forward. And the force which is resisting this, which is also a combination of friction and air resistance and whatever else, it's basically resisting this motion. So, if you're thinking that you are moving with a constant speed along a straight line, it doesn't mean that there are no forces. It just means that the force of the engine is neutralized by the force of the resistance, which is the friction and the air resistance, etc. So, the total, the balance of all forces is zero. Okay, so anyway, we're talking about the force as being a balance of all forces. Now, what if you press the gas pedal on the engine? I'm not talking about electric cars. Electric cars have their own equivalent of the gas pedal, right? Okay, so whenever you press the pedal, you increase the force of the engine with which it pushes the car forward, right? And car accelerates. The speed is increasing because the resisting forces like friction and resistance of the air probably are also increasing, but not as much. So, the balance of forces is positive and there is certain force which causes the acceleration. And that acceleration is also changing because sometimes you can press harder and you will move, you will increase your speed faster, let's say from 20 miles an hour to 40 miles an hour in like five seconds. But if you press it differently, then the same increase of the speed would be during a longer period or a shorter period of time, depending on how you press. So, basically I would like to extend this formula towards this type. Now, let's go to the mass. Is mass variable? Well, usually mass is not variable, but sometimes it is. Let's consider you are talking about a rocket. Now, when the rocket is going up, the fuel is burning, right? And the fuel has certain mass. Therefore, the mass of the rocket is changing. It's decreasing unsubstantially because the fuel represents a very large amount of mass of the rocket. So, all these variables can change. Okay, now, these are physical aspects. Now, what are the mathematical aspects? What is basically the goal of this particular equation? Most likely the goal is knowing the force even if it's dependent on time. And knowing the mass, again, even if it's dependent on time. We have to find out the position of our object. Now, what is the position? Well, we have acceleration, right? But acceleration is a second derivative of the position. So, what is this thing? It's a differential equation, ordinary differential equation, because we're talking about function of one argument, and it's a differential equation of the second order because it's a second derivative. In the previous lectures, we were talking only about the first order differential derivatives, different types, how to solve them, etc. And this is our first encounter with differential equation of the second order. Well, can it be solved? Before doing that, let's just consider two very important partial cases, very particular cases. Okay, now, let me just write down this equation on the top, and then I will go to particular cases. So, we have function of t is equal to mass of t times a, well, I will put, instead of a, I will put second derivative of position. Okay. So, where v is my first derivative, and a is derivative of the speed. Okay. Now, let's go to a particular case, and let's consider that f of t is equal to zero, all this. Again, it's not a single force, it's a balance of all forces, like in case of the car moving with constant speed, we are talking about the force of the engine is neutralized by the force of the resistance, friction and resistance of the air. So, what happens in this case? And let's also consider mass constant, just a constant, some kind of positive constant. So, what does it mean? Well, it means, which means second derivative of some function, which is distance as a function of time is equal to zero. Because mass is not equal to zero, right? Now, what if the function, if the second derivative is equal to zero? Well, the second derivative is the derivative of the first derivative, which means that my first derivative is constant, right? Remember this, if the derivative of a function is equal to zero, then the function itself is constant. And the function right now is actually the first derivative. Okay, fine. So, we have basically come up with this particular rule that the speed, this is the speed, is constant. So, if my function, if my force function of t is equal to zero, which means there are no external forces which are acting upon this object, or they are all neutralize each other, same thing. So, if all these forces neutralize each other and the result, the balance is zero, then my object would have a constant speed. See, whatever the speed is. Now, what kind of speed is this? Well, it depends. Because if the object already is moving with certain speed, and at some moment while it's moving with certain speed, we are saying that, okay, there are no more forces acting on this particular object. It will continue moving with that speed. Whatever that speed is, if it's a 20 mile an hour, it will be 20 miles an hour. If it's a 40, it will be 40. So, basically it all depends on the speed at certain moment, at which moment we actually can say that, okay, from this moment on, there are no forces acting. The most convenient way is actually assign the moment t is equal to zero, as the moment we start actually paying attention to our object. So, whatever the speed was at that particular moment, it's retained. And, again, going back to physics, this is basically the first law, Newton's. So, the first law of Newton says that if there are no forces, or the balance of all forces is equal to zero, acting upon an object. Well, it will basically continue doing what it's doing. If it was at rest, I mean, if its speed was zero, it will continue to be at rest if there are no more forces. If it was moving with certain constant speed, it will continue moving with that constant speed. So, basically we can say that if we know speed at moment t is equal to zero, let's say it's b zero. This particular speed will be maintained. Now, if this speed is maintained, how can we find out the distance from the beginning of the origin of coordinate? Okay, I will put this is equal to v zero, which is speed at moment zero. It's a constant. Now, if my first derivative is some kind of a constant, then we know that the function is what? We're just integrating, right? Integral, which is v zero times t plus some kind of other constant. Well, let me use different letter, constant g, right? We have to integrate it, because x t is integral of x prime t, right? So, integrating a constant gives me this plus some kind of a constant, which I don't know. Now, how can I basically find out what exactly this constant is? Same thing. We need some kind of initial condition, because all I have to know is where exactly my body was at moment zero. If my object was at point x zero at moment t is equal to zero, then this is exactly the constant I need. So, we need two initial conditions where exactly our object was at moment zero and what was its initial speed. If I know these two things, then knowing that the function which represents my force is equal to zero, then I can actually determine position of my object at any future moment of time. So, it's predetermined. So, if I know that there are no more forces acting upon this guy and masses, again, constant, I assume, then I have to know where exactly this particular object was at moment when I started my observation and what was its initial speed. If I know these two, then that's a position on my x axis where this particular object will be at any future moment of time. And again, let me just tell you again that my second law, my Newton's second law allows using these manipulations with differential equations to describe with the first law because this is actually the representation of the first Newton's first law, that if there are no more forces, then the body will continue its movement with whatever the initial speed was and whatever was the beginning position at the beginning of time. And if the initial speed was zero, then obviously this thing is zero and the position will not change this time. So, if the body was at rest when the zero was equal to zero, then it will remain at that particular position as it would change. If it was moving with some speed, it will continue moving with the same speed and proportionately at time the distance will increase from the initial distance to whatever it is. So, that's my first kind of application of this particular differential equation of the second order in case when there are no more forces. So, that's when the second Newton's law actually implies the first one. Okay, now let me go back to a little bit more common case, I would say, the force is not zero, but it's some kind of a constant. So, f is not zero, it's some kind of a p which is constant and m is m also constant. So, when my force is a constant and my mass is a constant, I have an equation of this type. Well, mass is not equal to zero, so I derive this and this. Integrating this, I will get, obviously, by the way, if f is a constant and m is a constant, then this is also a constant. So, I can also put it equals to a and also constant. Okay, so now what is my integral of the second derivative which is the first derivative? Integrating this constant would be a times t plus, plus, that's important. Plus, we have obviously some kind of a constant. Now, what is this particular constant? Same thing, if t is equal to zero, my speed must be something. So, I can safely put here v zero. Now, integrating this, now I don't have derivatives anymore. And this is function of t, obviously. Okay, integrating this, a is a constant, integral of t is t squared over two, right? Because the derivative of t squared is two t, divided by two would be t. And integral of v zero would be v zero times t plus constant. Or d, I don't know, I already used c, d. So, what is the constant? Again, we have to know the initial position. So, if t is equal to zero, initial position would be exactly x zero, right? So, we know this constant. So, now, this is an equation of the motion where my force is a constant, my mass is a constant. And therefore, my acceleration, a, is a constant. So, if the constant acceleration, then this is a quadratic polynomial, by the way, is an equation of the movement. So, you don't really have to remember these whenever you're studying physics. You don't really have to remember all these formulas. But you do have to know how to integrate, basically, that's it. And then you will derive this formula through a very simple integration process. Okay. Now, in general, so we have considered two particular cases. When force is equal to zero, and mass is a constant. The second case, when force is equal to some positive constant, and mass is still a constant. That's this one. Well, what is the general case? Well, general case when everything is changing, right? Well, when everything is changing, we can't really say anything about this. We can't say anything about mass. So, all we can say is that if we know how the force is changing with time, and mass is changing with the time. All we can say now that my second derivative is force divided by mass. My first derivative is equal to, as a function of t, of course, integral of f of t, m of t dt. And, well, obviously, there is a constant embedded here, right? Whenever I will take this integral, I will have to put plus some kind of a constant. So, my position at time t is another integral. Integral of integral of f of t divided by m of t dt and dt again. So, we integrate twice. First, we integrate. We get some kind of a function, plus constant, by the way. And the second integration gives us another function with another constant. Well, basically, that's what I wanted to talk about today. I just wanted you to understand that differential equations are not something which mathematicians just thought about and basically invented from within air. It's really going back to Newton, who was trying to express the laws of movement. And his laws were actually the reflection of differential equations. And he is one of the people who basically invented all these differentials, derivatives, integrals, et cetera. He, basically, one of those he and Leibniz were the pioneers in this particular era. And that, again, is going back to like 17th century or 18th, don't remember. Anyway, so these are the simplest differential equations, obviously. You see, it's just straight integration of this. It's not a more general. More general would be like function, some kind of function where the first derivative, the second derivative, the function itself, all are together. And that requires certain approach how to solve it. But these guys are solved very easily. And there is no problem approaching these particular solutions. And in the future, in my lectures in physics, I will definitely use all these differential equations and derivatives and integrals to basically go to the roots of all these physical formulas, which you might be presented without any kind of a proof. But there is a very simple mathematical proof, like the proof of the, whenever you have a constant acceleration, for instance, you have this quadratic polynomial, which express the position of the body. All right, so that's it. Thank you very much and good luck.