Transcript
00:00:230Paolo Guiotto: Okay, so… Last timer, Let me just quickly, review what,
00:14:970Paolo Guiotto: what we discussed the last time. So, we…
00:19:660Paolo Guiotto: reintroduce the concept of differential equation. You already know what is it, so do not,
00:27:120Paolo Guiotto: waste too much time. So we limit to so-called first-order equations, so where the derivative of the unknown is just the first derivative, and they have this particular form, Y prime equals FTY.
00:45:410Paolo Guiotto: It is very important, the function F, that determines the structure of the equation. So, the function F has a domain in the Cartesian plane of TY, so R2,
00:58:650Paolo Guiotto: That domain, in particular, is the domain where the solutions live, no? Because we need to compute F of TYT.
01:10:660Paolo Guiotto: We have also a geometrical interpretation of this quantity, because if you look at the equation, it says Y prime equals FTYT.
01:18:880Paolo Guiotto: So, this means that the value F of TYT gives the value of the derivative, the slope of the solution.
01:27:890Paolo Guiotto: This will be very important for what we are going to do. So you may imagine that knowing the value, so if you know that F at TYT is 1, it means that the slope of the solution is 45 degree increasing, and so on.
01:44:580Paolo Guiotto: So we noticed some phenomena here.
01:49:880Paolo Guiotto: And while for linear equations, we always have infinitely many solutions. This is a general fact for differential equations.
01:58:770Paolo Guiotto: But a unique solution when we impose an extra condition. Important extra condition is this one, the passage condition. So we want to find the solution of the equation such that at certain given time T0, the value of the solution is a given value, Y0. So T0, Y0 are known values.
02:19:510Paolo Guiotto: So it means to look for a solution that passed through that point. Now, for linear equations, we have seen, at least in an example, that the solution is always unique. It exists, and it is unique.
02:34:820Paolo Guiotto: For nonlinear equations, and moreover.
02:37:850Paolo Guiotto: All the solutions are defined for the same set of T, so they have the same lifetime. It's like if all people would live the same time, no? You mentioned…
02:49:470Paolo Guiotto: Well, for nonlinear equations, this is different. First of all, we have seen that the lifetime of the solution can be different, depending on the solution. Here, for this equation, we have solutions that are defined
03:05:210Paolo Guiotto: For forever, like the constant solution, the right one, or solutions that are defined from a certain time
03:13:740Paolo Guiotto: Up to plus infinity, or so forever in the future, or other solutions which are defined forever in the past.
03:22:60Paolo Guiotto: So these ones are defined forever in the past, and these ones are defined forever in the future. So it's not true that there is the same lifetime for the, all the solutions. This depends on the solution itself. So in particular, it depends on the initial condition.
03:43:200Paolo Guiotto: And that second example is even more disturbing, because it says that we could also have equations where, nonlinear equations.
03:55:390Paolo Guiotto: Where the same Cauchy problem has infinitely many solutions, which is really a big problem in real-world applications.
04:05:170Paolo Guiotto: So it is very important to understand, to ensure at least, that under good conditions on what? Of course, on the unique ingredient that describes the equation. The unique ingredient that describes the equation is the
04:20:640Paolo Guiotto: right-hand side of the equation? Is that function F? Because, well, the equation is Y prime, okay, Y prime, but what changed the equation is the function F.
04:32:240Paolo Guiotto: So if the function is linear, it… you have something. If function is nonlinear, you have something else, okay? So you see, it's the function f that determines what kind of equations we… we have in front.
04:45:680Paolo Guiotto: So what we want to see now is quickly, some I will just limit to one particular, which is the most important, most general result. One theorem that ensures that under good, good in the sense that easy to be verified conditions on that function F,
05:06:70Paolo Guiotto: We can say that a Cauchy problem has always
05:10:10Paolo Guiotto: just one and only one solution. So as one is usually said, existence, and only one is usually said uniqueness, okay? So that's why you hear his… hear these two words. So let's talk about, quickly, this…
05:29:430Paolo Guiotto: Which is called the Local… existence…
05:39:840Paolo Guiotto: And… uniqueness.
05:47:720Paolo Guiotto: There are several results, but, since we are going to use just this one, and we have not so much time, so we have a… we just focus on this one. So, latter, F…
06:02:150Paolo Guiotto: function of TY, be defined on a domain D of R2,
06:08:700Paolo Guiotto: real value. This is the functions that determine the equation. Notice that here, Y is Y, is the variable, it's a number, okay? It's not yet YT.
06:19:860Paolo Guiotto: So, we will use the same letter Y to… to…
06:25:150Paolo Guiotto: to denote two different things. So here, Y is the variable, and Y somewhere else, will be Y of T is a function, okay? But from the context, you should be able to distinguish which one is the right notation.
06:39:230Paolo Guiotto: So let F be a function, as usual, let's help with the intuition with the figure, so we have to imagine that we have a domain. I always draw this domain as if it contains the origin, it is bounded, but of course, it can be whatever, okay? It's just to have an idea.
06:58:550Paolo Guiotto: Okay, so, We have this, theorem.
07:08:270Paolo Guiotto: So, assume that.
07:15:170Paolo Guiotto: The goal is to give existence and uniqueness of a solution of the Cauchy problem. Let's write here, I will repeat down here. The Cauchy problem will be a differential equation, Y prime equal FTY, plus a passage condition, YT0.
07:35:390Paolo Guiotto: Y0. We want to see, under which conditions on F, we can ensure that this problem has a solution and just one solution.
07:44:630Paolo Guiotto: So assume that the two requirements are, let's say, more or less easy requirements for F. The number one is very easy, because it is… the function f is just a continuous function on its…
07:58:120Paolo Guiotto: domain. So, as you can see, it's not such a big restriction.
08:03:450Paolo Guiotto: But this condition is, for example, verified also in the case we considered the last example of last class, where we
08:13:260Paolo Guiotto: I've seen an example of non-uniqueness. That function, the cubic root of y, as function of T is constant, so it is continuous. As function of Y is the root, root is continuous.
08:27:250Paolo Guiotto: Maybe it's not differentiable, but it is continuous. So, this assumption is verified here. So, alone is not sufficient. And in fact, there is a second assumption. The second assumption is that we want that the derivative of that function f with respect to just Y
08:45:850Paolo Guiotto: Dob.
08:47:590Paolo Guiotto: Derivative with respect to Y of the function f also continuous on the main D.
08:53:890Paolo Guiotto: And as you can see, in this example.
08:57:380Paolo Guiotto: In this bad example, this assumption is not verified.
09:01:730Paolo Guiotto: Because if you would compute here the derivative with respect to Y of this F… now, this F is Y to 1 3rd, so the derivative with respect to Y would be 1 3rd Y to minus 1 third minus 1 is minus 2 thirds.
09:18:830Paolo Guiotto: And what happens here? That when Y is 0, the function is defined, but the derivative is not.
09:24:350Paolo Guiotto: the derivative explodes, so there is not derivative, there is not even the derivative, okay, at points E0.
09:35:880Paolo Guiotto: they are in the domain, because the domain here is at 2. So the derivative is not even defined, and therefore it cannot be continuous. So you see that the why this example has this phenomenon, because
09:50:380Paolo Guiotto: When you have the partial derivative with respect to Y, well-defined and continuous.
09:57:60Paolo Guiotto: As you will see, we get the conclusion about existence and uniqueness.
10:01:600Paolo Guiotto: Now, this second condition is also easy to be verified, because we say continuity is something that we discuss
10:08:930Paolo Guiotto: normally, except for particular cases, we discuss easily, no? So, it's easy to check that an F is continuous, and the derivative with respect to Y of that F is also continuous. So, we can say that
10:23:560Paolo Guiotto: It's a very, very, let's say, mild assumption.
10:29:530Paolo Guiotto: Okay, then… What this theorem says is.
10:34:870Paolo Guiotto: So, first of all, about the Goshi problem, since we looked for a solution that passes to that point, so the point with abshysa T0 ordinate Y0. As we said, this is a point of the plot of Y, no?
10:53:440Paolo Guiotto: So it must be in the domain. So this means that you cannot… you can never take an initial condition outside of D, okay? So this theorem says, then.
11:03:40Paolo Guiotto: for every initial condition T0Y0 in D,
11:12:300Paolo Guiotto: is not a restriction, I repeat. You cannot take an initial condition outside of the… that's a nonsense, so it's not a restriction.
11:20:810Paolo Guiotto: There exists a union.
11:27:600Paolo Guiotto: Why?
11:30:210Paolo Guiotto: defined on an interval that we will do not, with this standard notation, alpha, beta, open.
11:37:970Paolo Guiotto: That's important. Real valid.
11:41:280Paolo Guiotto: solution, off, because she…
11:49:170Paolo Guiotto: problem.
11:51:940Paolo Guiotto: So let's copy Y prime equal FEY.
11:57:720Paolo Guiotto: with the passage condition YT0 equals Y0.
12:09:30Paolo Guiotto: well, let's say this better. There exist… well, I modified this, I will write unique later. There exist, then, there exist.
12:24:140Paolo Guiotto: Solution, Y, solution of the Cauchy problem.
12:28:490Paolo Guiotto: Such that, huh?
12:32:200Paolo Guiotto: And this is the sense of uniqueness. I will help your intuition with the figure. So this solution is the solution… is a solution of the differential equation that passed through that point.
12:47:480Paolo Guiotto: This solution has the following characteristics, that whenever you get another solution.
12:54:780Paolo Guiotto: of the Senkoshi problem, such that if Well, that's,
13:05:310Paolo Guiotto: Such that, let's give a name, Uniqueness, to this part.
13:13:370Paolo Guiotto: If, let's say Y tilde.
13:17:630Paolo Guiotto: is defined… it's important, it's detailed, but it's important. Defined on some interval, i, real value.
13:28:280Paolo Guiotto: is another Is, well, let's say, any other
13:37:590Paolo Guiotto: solution.
13:40:80Paolo Guiotto: off.
13:41:80Paolo Guiotto: It's important. D, same, not of nada, the same Cauchy problem.
13:52:470Paolo Guiotto: So, you have a second solution, let's say… The green one.
13:59:350Paolo Guiotto: a solution of the differential equation that passed through that point. And what should happen?
14:05:300Paolo Guiotto: Well, it should happen that the idea is that the two, the green and the black, must coincide.
14:13:840Paolo Guiotto: But, there is an important addition to do. Then… Number 1!
14:21:920Paolo Guiotto: The interval of definition of the other solution is necessarily a subset of the interval of definition of the solution granted from this theorem.
14:36:470Paolo Guiotto: Okay? So, this means that you cannot have any other solution of the Cauchy problem that is defined
14:44:130Paolo Guiotto: beyond where this solution is defined, beyond in the future or in the past, because the time interval of definition must be contained there, okay? And second, the two must coincide, so the Y must be identically equal to Y tilde where? Well.
15:03:690Paolo Guiotto: Y is defined on alpha-beta. Y tilde is on i, which is not necessarily the same, but can be different, but because of this condition, it's smaller, so on the common interval, which is the interval I.
15:19:920Paolo Guiotto: Okay? So if you look at this figure, it means that… well, let's do a second figure here.
15:27:330Paolo Guiotto: to show the possibilities. Imagine, this is domain D, this is our initial condition, this is the solution Y.
15:35:830Paolo Guiotto: Now, can I have a solution Y tilde like that? So, solution that passed through the same… no.
15:44:150Paolo Guiotto: Because this is impossible.
15:48:780Paolo Guiotto: Because, you see, this green line
15:52:20Paolo Guiotto: which is defined on this interval, this is the interval I on the axis where the green solution is defined. On that interval, does not verify the second condition, does not coincide with the Y, so this is impossible.
16:11:570Paolo Guiotto: Can I have this, huh?
16:14:980Paolo Guiotto: still domain D, the passage condition. I have my solution Y, and the second solution which does this.
16:25:560Paolo Guiotto: So, you see, the black solution is my Y.
16:33:370Paolo Guiotto: This is the passage point, T0.
16:36:860Paolo Guiotto: Y0, and the black line is Y, the green line is Y in that.
16:42:680Paolo Guiotto: This cannot be possible, because this would mean that the interval of definition of the Y tilde is not contained in the interval of definition of the Y. If this is the figure, this is the interval alpha, beta.
16:58:660Paolo Guiotto: So the solution Y is defined here, in the blue interval, and the green solution would be defined in the yellow interval.
17:10:10Paolo Guiotto: So you see, in this case, I wouldn't be contained into alpha-beta.
17:17:380Paolo Guiotto: So, also, this figure is impossible.
17:23:480Paolo Guiotto: Okay.
17:24:530Paolo Guiotto: So, this theorem says that under this mild condition, F and the DYF both continuous on the domain, whatever is the initial condition we take in the domain D, there exists a solution of the Cauchy problem.
17:39:180Paolo Guiotto: And that solution is unique in the following sense, that you cannot construct any other solution that is alive for a longer time than the solution Y.
17:50:750Paolo Guiotto: So that's the… in fact, it is called the maximal solution, because you cannot extend far the life of this solution.
18:01:20Paolo Guiotto: Okay.
18:02:40Paolo Guiotto: Now, the key point is that this theorem
18:05:720Paolo Guiotto: It's like Viasta's theorem, basically. Actually, it's not a… it's a constructive theorem. In fact, the proofs gives also an algorithm that is actually used to build numerical solutions, okay? But since this is not a course of numerical analysis, you will see numerical calculus
18:24:160Paolo Guiotto: you maybe… I don't know if you… I'm sure that sooner or later, you will do some numerical analysis, and a classical problem is to build solutions of differential equations, to build algorithms, to
18:38:790Paolo Guiotto: To make, possible for a computer to… to… to compute numerically a solution.
18:45:380Paolo Guiotto: Why do you need to compute numerically? Because here we are considering the case where we cannot solve the equation, so we cannot find exactly the Y.
18:55:640Paolo Guiotto: But let's say, apart from this story that we forget, because we do not touch this question of numerical
19:02:110Paolo Guiotto: methods. From the point of view of the user, this theorem is like the Beister's theorem. The Beister's theorem says you have a continuous function on a closer than bounded interval, or on a compact set. This function has both minimum and maximum. Yes, but how do I find minimum and maximum?
19:22:320Paolo Guiotto: No answers. The Weissler's theorem is a pure existence theorem, and that one is similar, because it says, whatever is the initial condition, you are sure that there is a unique solution to the Cauchy problem. Yes, but how can I find this?
19:36:700Paolo Guiotto: So, now here we have to reflect, to think about a second to what does it mean to solve that problem? Because,
19:45:310Paolo Guiotto: Normally, in school, we learn that to solve means exactly one thing. To find, for example, here, Y equals this.
19:56:650Paolo Guiotto: Now, you have to solve an equation x equals this value. You want a formula, no?
20:02:740Paolo Guiotto: But that's a particular way to say we solved the problem. Now, in the contemporary science.
20:10:460Paolo Guiotto: Solver means also to be able to write an algorithm for a computer that is able to plot the solution, or to compute numerically. I want to know what is the value, the numerical value of that solution tomorrow, I can do.
20:27:250Paolo Guiotto: It's the machine that does for me, but I give instructions to the machine to do this calculation.
20:34:630Paolo Guiotto: This is acceptable as solution exactly for practical purposes, no? If you are a real-world user.
20:41:770Paolo Guiotto: you do not make such much difference between the abstract, exact formula and a numerical formula, because at the end of the day, you need to compute values, so you need numerical values. Even if you have an exact formula, you will plug into a computer and you will do calculations to get the exact values, so there is no difference from practical purposes.
21:03:870Paolo Guiotto: So there is a… Also, the idea of solving numerically an equation, which is perfectly fine.
21:12:190Paolo Guiotto: Now, this is not the course of numerical analysis, so we do not touch these topics, which are themselves in theory. It's not just a question of do
21:22:920Paolo Guiotto: toward the applications of what we have seen. It's a… it's a discipline, an interdiscipline, a scientific discipline, numerical analysis.
21:33:620Paolo Guiotto: So what can we do here?
21:36:80Paolo Guiotto: Now, what can we do here is to use our tools to add some idea on the solution of the equation without solving the equation.
21:48:520Paolo Guiotto: So, the point is that,
21:51:570Paolo Guiotto: So, suppose that you have a function.
21:54:510Paolo Guiotto: Last year.
21:57:520Paolo Guiotto: I'm pretty sure that you have done several times this kind of exercise. You have a given function, and you want to plot the graph of the function to see how this function is made, no?
22:10:930Paolo Guiotto: Of course, you plot a qualitative graph, you do not plot an exact graph, you do not do any explicit calculation, or you compute a few points, but you do not compute x by x the value of f of x, right?
22:24:970Paolo Guiotto: That exercise provides you an idea on how that function is made, and that is interesting, because qualitatively, if you look at the graph of the function, you can see where there is the minimum, the maximum, what the function does, if it grows, if it is decreasing, and so on.
22:42:130Paolo Guiotto: So, what we want to do here now is exactly the following thing.
22:46:610Paolo Guiotto: Can we provide the plot of the graph of the solution, but without solving the equation? Because clearly, if I solve the equation, I have a function, then I use the tools I learned last year to plot the diagram of the function.
23:03:730Paolo Guiotto: So I would say. You want to see the plot of the solution, solve the equation, step one. Step two, you have a function, then you study that function, and you produce a graph of the function. You have your plot.
23:19:190Paolo Guiotto: But if we cannot do the step number one, so we cannot solve the equation, how can we plot the solution without the solution explicitly in n?
23:29:250Paolo Guiotto: So is that possible? I want to convince you that with basically a few tools, we can do that kind of job, okay? So this problem…
23:40:660Paolo Guiotto: It's called the, problem of, qualitative…
23:51:290Paolo Guiotto: plot.
23:53:650Paolo Guiotto: of… solutions.
23:57:400Paolo Guiotto: Of a cushy problem.
24:01:990Paolo Guiotto: So… The problem we want to solve is the following.
24:06:40Paolo Guiotto: So we consider…
24:12:920Paolo Guiotto: Cause sheep problem.
24:14:840Paolo Guiotto: So… the equation, Y prime equals FT.
24:19:150Paolo Guiotto: Why?
24:20:510Paolo Guiotto: with some passage condition, YT0 equals Y0. So, of course, we assume that in concrete examples, as you will see, T0, Y0 are known.
24:31:710Paolo Guiotto: We assume…
24:37:460Paolo Guiotto: that… F verifies… assumptions.
24:53:500Paolo Guiotto: of, local… existence.
24:59:640Paolo Guiotto: And… uniqueness.
25:06:340Paolo Guiotto: on B.
25:07:980Paolo Guiotto: That means F and DYF are continuous on the…
25:14:480Paolo Guiotto: So the assumptions of the theorem in just take.
25:17:510Paolo Guiotto: Now, the problem is,
25:23:220Paolo Guiotto: Is it possible…
25:29:20Paolo Guiotto: to lots.
25:32:260Paolo Guiotto: the… graph.
25:37:180Paolo Guiotto: off.
25:38:750Paolo Guiotto: YFT?
25:40:800Paolo Guiotto: without…
25:48:770Paolo Guiotto: solving… Well, explicitly solving
26:03:740Paolo Guiotto: Was she problem?
26:06:600Paolo Guiotto: So that's important.
26:08:640Paolo Guiotto: We want to plot the graph without solving, basically, the equation, because if you…
26:15:710Paolo Guiotto: If you solve the equation, then the solution of the Cauchy problem means determining that arbitrary parameter in such a way that you verify the pass condition. So clearly, the main problem is solving any equation, but can we do that kind of argument?
26:31:600Paolo Guiotto: Now, the point is that I want to convince you that we can do that. There is not a general theory. It's just a matter of, gather the… what we know.
26:42:800Paolo Guiotto: from properties of one variable function, so from first-year calculus, basically, mostly it is this one. So I will illustrate, this.
26:54:760Paolo Guiotto: on, on some examples first.
27:00:410Paolo Guiotto: Let me… let me… one second…
27:06:100Paolo Guiotto: Okay, so let's take one of the… to begin.
27:11:60Paolo Guiotto: Exam… example.
27:13:850Paolo Guiotto: Let's, imagine that we want to plot D.
27:20:00Paolo Guiotto: solution.
27:21:500Paolo Guiotto: of this Cauchy problem. We take any question for which we know the solution, okay?
27:27:520Paolo Guiotto: So Y prime equals Y squared, we have seen last time, we solved, but now we do not touch the solution. We want to arrive to plot the solution without solving the equation. By the way, let me just refresh you that it is this problem here.
27:45:340Paolo Guiotto: You see?
27:47:960Paolo Guiotto: Y prime equals Y squared, YT0 equals Y0. Now, I will take a specific T0, Y0 to make easier the discussion, but
27:56:680Paolo Guiotto: the equation is the same, so we have seen that the equation can be solved, we have seen the solution at this, etc, etc. But we forget of all this. We want to try to
28:08:00Paolo Guiotto: Have, at the end, a figure similar to these ones.
28:12:390Paolo Guiotto: For example, this one, without computing the solution. The question is, is that possible?
28:18:450Paolo Guiotto: Okay, let's see what can be done.
28:21:60Paolo Guiotto: So, we fix… to fix ideas, we take a Cauchy problem, Y0 equals 1. Just to make you, the idea, Y0 equals 1 is… we said that in this problem that E0 is irrelevant, what change, the situation is Y0 positive, Y0 negative.
28:40:70Paolo Guiotto: So T0 is 0, means that the condition is given at time t0 plus 0, and that the Y0 is positive. So we are exactly in this case, no? So the solution should be a solution that does something like this, okay, with an explosion somewhere.
28:56:360Paolo Guiotto: Now, the point is, can we, can we obtain, re-obtain this figure? Yes, no? Let's see.
29:05:310Paolo Guiotto: Okay, so consider this.
29:07:950Paolo Guiotto: Now, let's start a little bit,
29:11:100Paolo Guiotto: without a precise… a precise way, so let's see what can be done. First of all, let's analyze the equation. This is the FTY.
29:24:300Paolo Guiotto: So, here…
29:29:200Paolo Guiotto: TY is Y squared.
29:32:760Paolo Guiotto: is defined… Remind that the domain D is the set of TYs.
29:42:120Paolo Guiotto: for which FTY makes sense is, well… Well, defined.
29:52:40Paolo Guiotto: In this case, there is no condition on T, there is no condition on Y, so this domain is just R2, okay?
30:00:900Paolo Guiotto: So… It's, in this kind of problems, it's a very important factor to start drawing.
30:09:270Paolo Guiotto: things. Because each information you get from this analysis, it's a qualitative information that you should put into a figure.
30:18:730Paolo Guiotto: First of all, to understand what you are doing, and exactly for the same reason, you have with the problems when you study a function, the most important error you can do is a coherence error.
30:33:870Paolo Guiotto: So you may, for example, you are getting the derivative, the function is increasing, but the limit is minus infinity. How is that possible? That's an incerence between the limit and the sign of the derivative. One of the two is wrong.
30:48:270Paolo Guiotto: Maybe both, okay? But certainly, that's not possible. So, how can you recognize that there is a problem?
30:56:170Paolo Guiotto: In an exercise where you have to gather many informations, no? Limits, computing derivatives, solving inequalities, interpreting results, computing second derivatives for the concavity, etc. All this together must produce a coherent final figure.
31:15:140Paolo Guiotto: So you have a lot of small problems, different in nature, that should stay together to give a current picture. It's very easy to a mistake there.
31:25:900Paolo Guiotto: Okay? And the unique way you have to understand that you're doing a mistake is just to put everything in a figure, step by step. So do not wait until the end to do a figure, but do the figure since the beginning. So this is our plain TY, and for the moment, the domain is everything.
31:45:530Paolo Guiotto: Okay.
31:46:660Paolo Guiotto: So, let's see some standard and important, as you will see, things that can be said. So, first of all, is BCEF verifying assumptions of local… ensuring local existence and uniqueness? So, number one.
32:03:600Paolo Guiotto: So, let's say… Is F verifying
32:12:410Paolo Guiotto: existence, and… Uniqueness, Assumptions?
32:21:00Paolo Guiotto: Well, what are these assumptions? They're not particularly demanding, because F continues, DYF continues, and that's true, because
32:30:140Paolo Guiotto: Clearly, the answer is yes.
32:34:450Paolo Guiotto: FTY is Y squared, is clearly a continuous function on R2,
32:41:750Paolo Guiotto: It doesn't matter that you see one variable. That's because of this case that is not T, but this means it is constant in T. It's not a function of one variable. It's a function of two variables, but there is no T means it's constant in T. It's not… it's not a function of T, okay?
32:58:250Paolo Guiotto: And DYF, it's easy, it's 2Y, they are both polynomials, so they are both continuous in R2.
33:06:890Paolo Guiotto: So… that stops.
33:10:390Paolo Guiotto: Number two.
33:11:960Paolo Guiotto: Another special, which is not important for the Cauchy problem, apparently, here.
33:18:720Paolo Guiotto: Another important information that you can draw from any question of this type is looking for very particular, very spatial solutions, easy solutions. The easiest possible solutions are constant, okay?
33:34:730Paolo Guiotto: So, constant…
33:40:260Paolo Guiotto: They will play an important role, as you will see. Solutions.
33:46:60Paolo Guiotto: So… are dead.
33:52:730Paolo Guiotto: So…
33:55:290Paolo Guiotto: I'm saying we solved the equation, but we do not solve in general the equation. Now, solving for constant solutions is… can be done normally in whatever is the equation.
34:07:240Paolo Guiotto: Solving the equation for searching for all the possible solutions cannot be done in general.
34:15:440Paolo Guiotto: For example, in this case, could be done because it's a separable variable equation, but if the equation is not of that type, you do not have a method, okay? While, for constant solutions, you can always do, because, think about Y constantly equal to C, let's say, is a solution
34:33:820Paolo Guiotto: If and if… well, when you plug this into the equation, the equation is verified. The equation is Y prime equals Y squared. What is the derivative of, this Y?
34:49:239Paolo Guiotto: Y is constant? So the derivative is…
34:52:199Paolo Guiotto: 0. So, the equation becomes left-hand side 0, right-hand side C squared. So, it becomes an algebraic equation, it's not a differential equation. You have to look for the value of the constancy, if any.
35:08:620Paolo Guiotto: for which this is verified, and that's easy, because we get C equals 0 in this case. So…
35:15:520Paolo Guiotto: Yes, sir?
35:17:970Paolo Guiotto: degrees… Just… one constant.
35:26:40Paolo Guiotto: solution Y of T identically equal to 0. And that's important also to plot the constant solutions, because they will be very important here. So let's go back to this figure.
35:39:720Paolo Guiotto: And now let's plot with this red…
35:43:770Paolo Guiotto: the constant solution, so that's Y constant equal to zero.
35:47:690Paolo Guiotto: Is that the solution we are looking for? No, because our solution verifies Y of 0 equals 1. So let's also add the passage point, which is this one.
36:01:70Paolo Guiotto: So the solution must pass through that point, green point. So it certainly cannot be the red solution, okay? So why we did that? As you will see, it makes sense.
36:12:240Paolo Guiotto: Another general fact that could be, and here we start entering in the, more precisely in the qualitative study.
36:24:200Paolo Guiotto: Is, can we determine where solutions are increasing, decreasing?
36:30:820Paolo Guiotto: You know, at the end, we have to produce a plot.
36:34:210Paolo Guiotto: So we have to understand if the solution is increasing or decreasing, something like this, okay?
36:40:170Paolo Guiotto: So, can we understand if solutions can be decreasing, increasing?
36:44:900Paolo Guiotto: wealth.
36:46:290Paolo Guiotto: Let's say,
36:52:470Paolo Guiotto: let's say, increasing… But let's say… let's formulate this, regions… of the domain T, Where… solutions.
37:10:730Paolo Guiotto: R… increasing…
37:17:130Paolo Guiotto: or decreasing.
37:22:590Paolo Guiotto: What does it mean, this?
37:24:460Paolo Guiotto: Now, I know this is a general point, so let's open a little parenthesis here.
37:31:710Paolo Guiotto: So, since the equation is Y prime, now I will restore the letter T for the solution for a second, okay? Y prime of t is F of TYT.
37:47:980Paolo Guiotto: You see that this is general, does not depend on this particular equation.
37:52:650Paolo Guiotto: You see that when the solution Y as function of t is increasing, we know that Y is increasing if and only if
38:05:00Paolo Guiotto: The derivative… ease?
38:08:750Paolo Guiotto: Pause.
38:09:940Paolo Guiotto: But the derivative is the right-hand side, coincide, because of the equation. So, if and only if FTYT
38:19:390Paolo Guiotto: Is greater or equal than zero.
38:23:250Paolo Guiotto: So this means, if and only if, let's give a different interpretation of this.
38:29:360Paolo Guiotto: You see, F evaluated at point TYT,
38:33:690Paolo Guiotto: Which is a special point, because it's a point of the graph, of the solution. Points T, YT are exactly the points that belongs to the graph of the solution. So, the point T, YT,
38:46:220Paolo Guiotto: must be in a special set, a set of points TY, well, the function f is positive.
38:57:670Paolo Guiotto: The point is that this thing is a subset of the plane, it's, in fact, a subset of the domain.
39:07:40Paolo Guiotto: So it's a region of D where, when the point TYT, so the point of the graph.
39:15:70Paolo Guiotto: Passes through that region, we are sure that the solution is going up.
39:20:490Paolo Guiotto: Okay? Now, look at this point. This is a subset, there is no, so this is a…
39:26:770Paolo Guiotto: you have to be careful with the reading of this YT. Here, YT is the solution, okay? And this means the point of coordinates PY of T,
39:38:60Paolo Guiotto: which is the point at time t on the graph of the solution, must be in this set, which is a set of points in the Cartesian plane, we still use the letter TY,
39:50:700Paolo Guiotto: Because, differently would be Even worse, to use a different letter.
39:55:850Paolo Guiotto: But now Y here is just the coordinates, no? These are the points TY in the Cartesian plane.
40:02:440Paolo Guiotto: Such that F is positive. So, they must be in the domain D, because F must be computed, and positive. So, this is a part of the domain, not necessarily all the domain, because maybe
40:13:130Paolo Guiotto: At some point, F is positive, at some other point, F is negative, no? So the domain will be made by all possibilities, and here we are taking only one possibility, where F is positive.
40:25:340Paolo Guiotto: Now, that region is the region where…
40:28:370Paolo Guiotto: When the solution is in this region, we are sure that the solution is going up, because in that region, F of TYT is positive. This means, because of the equation, Y prime greater or equal than 0. In our example.
40:52:230Paolo Guiotto: We can say that Y is increasing if and only if
40:58:420Paolo Guiotto: Y prime of t is positive.
41:01:210Paolo Guiotto: But this is F of TYT, which is, for our case, Y of T squared.
41:10:240Paolo Guiotto: Okay, so Y is increasing if and only if Y of T squared is positive, when this is
41:19:720Paolo Guiotto: So let me clean up YT square positive.
41:26:210Paolo Guiotto: For every T, doesn't matter what is T. For every T, for which…
41:35:350Paolo Guiotto: Why is, defined.
41:41:780Paolo Guiotto: Okay.
41:42:930Paolo Guiotto: So, it means that our solution, whatever it is, is always increasing.
41:49:50Paolo Guiotto: So, returning back to this figure, what can be said? It can be said that wherever it is the point TWT, suppose that at some time t, the solution passes through this point.
42:01:470Paolo Guiotto: Here, the solution will be increasing, so I will see something like that. I cannot see a solution doing this.
42:08:610Paolo Guiotto: Because this would be a decreasing solution. This is impossible, okay? So, we can say, a useful notation could be the following, put arrows on the plane, because here, but also here, solutions are always going up, growing.
42:26:310Paolo Guiotto: So I now can start imaging something about my solution I'm looking for. I know that my solution exists because we verify the existence and uniqueness.
42:37:920Paolo Guiotto: I know that something that passed through that point exists, and it should be an increasing thing, so I can start saying that my solution will do something like that.
42:49:880Paolo Guiotto: But then, now, the point is, okay, what can be, what can be said about,
42:57:00Paolo Guiotto: So let's say that this is alpha and this is beta. Can we say anything about these alpha and beta?
43:08:20Paolo Guiotto: Now, what I want to know is, is it possible that the solution dies there?
43:14:770Paolo Guiotto: So, that alpha is here, and then that's over, the solution, that's over… that's the… the… the time when the solution, was born, actually.
43:33:90Paolo Guiotto: Well, let's wait a second, but before we go there, let's say that for the moment, we know that the solution is increasing. Well, it could… if I go back to the past, I see my solution going down, if I go to the left. So, it could go down until where?
43:52:490Paolo Guiotto: Does it end somewhere here? Does it go down until the axis can go down, like this one?
43:59:340Paolo Guiotto: What happens, really?
44:01:770Paolo Guiotto: So, you see, we need now to know what happens when we go back to the past, and what happens when I go… I move forward in the future.
44:10:150Paolo Guiotto: can be this, the solution blows up, the solution maybe changes the concavity, and thus… there are many questions still open. But, for example, let's see that we can solve some of them.
44:24:280Paolo Guiotto: Okay, so we deduced that, in particular, our solution is, increasing.
44:38:350Paolo Guiotto: On its interval of definition, alpha-beta.
44:44:110Paolo Guiotto: interval of definition.
44:48:460Paolo Guiotto: of Y.
44:52:460Paolo Guiotto: Let's, for example, try to understand if our solution has the concavity, if it is concave or convex.
45:02:660Paolo Guiotto: See if the concavity is pointing upward or downward.
45:06:760Paolo Guiotto: So, let's discuss the concavity. Number… So, 3, number 4.
45:15:920Paolo Guiotto: Can we determine the concavity of why.
45:24:780Paolo Guiotto: Now, the concavity is determined by studying what?
45:29:640Paolo Guiotto: A second derivative. So we know that, for example, Y is, with the concavity pointing upward, technically we say Y convex.
45:41:480Paolo Guiotto: If, no, if… Why second is… What?
45:47:670Paolo Guiotto: Positing.
45:49:300Paolo Guiotto: Okay, what is Y second? We do not have Y sec… we do not even have Y for the moment, we have only the equation, and that's the goal that we have, no? So, we know…
46:03:610Paolo Guiotto: that Y prime is equal to Y squared, right? That's the equation.
46:09:50Paolo Guiotto: So, how can I get way second?
46:15:470Paolo Guiotto: Yeah, that's,
46:17:260Paolo Guiotto: the classical, let's say, hard work we have to do, no? You take this identity and you differentiate, no? If you differentiate the left-hand side, you get Y second.
46:30:550Paolo Guiotto: Of course, remind that Y here is a function of dia.
46:34:400Paolo Guiotto: So, if you want, you can write Y prime of t equal y of t squared. You understand that it's a function of T, I can't differentiate, no? Again, but let's keep the light notation. Why second? What would you write at the right-hand side?
46:53:600Paolo Guiotto: What?
46:55:980Paolo Guiotto: No. 2Y?
47:01:810Paolo Guiotto: times Y prime.
47:03:620Paolo Guiotto: That's the chain rule, no? If you derive Y square, you get 2Y, which is the derivative of the square, times the derivative of what is in the square, which is not the variable. The variable is T.
47:16:140Paolo Guiotto: Here, in the square, you have Y of T, it's a function, no? So, how would you derive, I don't know, sine of t squared? What is the derivative of this? How do you do?
47:26:870Paolo Guiotto: You do 2 sine T, and then you do not stop here.
47:31:480Paolo Guiotto: Because it's the square of sine t, and it's not the square of t. So you have to differentiate over sine, and you get cos T, no? That's the chain rule. That's very important, because if you forget of Y prime.
47:45:170Paolo Guiotto: Of course, you change the story here. Okay, now we know why second is 2YY prime.
47:52:800Paolo Guiotto: What do we know about these quantities? We remind that goal is way second positive, say, if way second is positive.
48:00:780Paolo Guiotto: What do I know? Well, what I know for sure is… What? You know the lifetime?
48:09:880Paolo Guiotto: Yeah, that's the true information we have. This is positive.
48:17:150Paolo Guiotto: And we don't… because we proved one minute ago, okay?
48:20:970Paolo Guiotto: And what about Y?
48:24:370Paolo Guiotto: It's positive, it's negative. Look at the figure. The figure would suggest that, at least initially, it's positive.
48:33:120Paolo Guiotto: But can become negative, so can this happen?
48:46:710Paolo Guiotto: Yes, and then?
48:49:880Paolo Guiotto: Yes, of course, there is something here very important to observe, and this justifies why the constant solution is very important.
48:59:150Paolo Guiotto: Because, as you say, if we want to become negative, sooner or later, we have to cross zero.
49:06:260Paolo Guiotto: And there, what do you see?
49:09:540Paolo Guiotto: You see something that is forbidden for solutions.
49:14:490Paolo Guiotto: They can never cross, okay? If you have uniqueness.
49:18:610Paolo Guiotto: Okay? That's why it's important having uniqueness. Now, let's see the precise argument that shows why this figure is impossible. So, since the solution is initially positive, it will stay all time positive. So, here we have a claim.
49:38:20Paolo Guiotto: So… Please, ma'am.
49:43:620Paolo Guiotto: I claim that Y of T is positive for every time t in the lifetime interval alpha-beta. Let's see why. Now, the intuitive argument is this one, what we explained.
50:00:650Paolo Guiotto: If the solution becomes negative somewhere.
50:05:500Paolo Guiotto: By continuity, it must be zero somewhere else, no? Because if you have positive here, you are negative here, there is a theorem which is called intermediate values theorem. You have zero somewhere in the middle.
50:20:440Paolo Guiotto: And that point's zero. What happens here?
50:23:920Paolo Guiotto: That if we consider the Cauchy problem.
50:27:630Paolo Guiotto: starting at this point, not at that point, but this blue point, for this Cauchy problem, I would have a solution, the green solution of the equation, and a red solution, another solution of the equation. So, two solutions of the same equation.
50:46:560Paolo Guiotto: For the same cushy problem.
50:48:490Paolo Guiotto: And the Bushik government says that the two solutions must coincide.
50:53:340Paolo Guiotto: Where they're both defined.
50:55:290Paolo Guiotto: The red one is defined forever. The green one, we don't know where is it defined. So, certainly, the alpha-beta is less than the real line. So, in particular, the green must coincide with the red, where the green nexus.
51:12:640Paolo Guiotto: But then, you see that the figure says that that's not the case, but then, at time t equals 0, they are both defined. The red is 0, the green is 1.
51:22:650Paolo Guiotto: And that's the contradiction, okay? Now, we have to write formally this idea. However, the key point is that two different solutions, when you have existence and uniqueness.
51:35:810Paolo Guiotto: That's why it's important. Can never cross.
51:38:510Paolo Guiotto: So let's write this, then we do the bracket.
51:41:380Paolo Guiotto: Okay, so… We know…
51:47:670Paolo Guiotto: Y of 0 is 1, so it is positive.
51:52:320Paolo Guiotto: So, if this claim, star, is false, so it's a contradiction argument.
52:02:540Paolo Guiotto: Then, there exists some time, I don't know, where is it? In the figure. Well, let's do the figure down here, okay? So, very quickly, the constant solution
52:13:890Paolo Guiotto: my solution, here is one. Suppose that I have a time
52:19:790Paolo Guiotto: T1, where the solution is negative.
52:23:760Paolo Guiotto: If there exists a time T1 such that YT1 is negative.
52:30:440Paolo Guiotto: Then, there exists also a time T2, such that YT2 is 0.
52:38:430Paolo Guiotto: And that's T2.
52:42:320Paolo Guiotto: by continuity. This is… by continuity.
52:47:390Paolo Guiotto: who says that Y is continuous?
52:51:370Paolo Guiotto: Why Y is continuous? Because Y is…
53:00:150Paolo Guiotto: Who says that it is a polynom? No? Why is the solution of the differential equation so? Why is
53:08:360Paolo Guiotto: differentiable. So there is derivative, if there is derivative, the function is continuous. Okay, by continuity, there is a time where the solution is zero. Okay, but then…
53:19:410Paolo Guiotto: So, the red solution, Y constant equal to 0, and the green solution And, our Y…
53:32:510Paolo Guiotto: of tea.
53:35:380Paolo Guiotto: Would be…
53:41:750Paolo Guiotto: both.
53:43:910Paolo Guiotto: solutions… of the Cauchy problem, So, equation, they are both solutions of the equation.
53:52:790Paolo Guiotto: With initial condition, with… Passage condition, why?
54:01:920Paolo Guiotto: at time, T2 equals 0.
54:06:160Paolo Guiotto: And then, since we have uniqueness.
54:13:200Paolo Guiotto: This comes from the check we have done initially. We get that the two must coincide where they are both defined on the smaller interval, which is alpha-beta. So, our green
54:26:330Paolo Guiotto: must coincide with the zero solution for every time T in
54:34:40Paolo Guiotto: alpha-beta. I say this one because the red one is defined forever, so both are defined on this.
54:41:100Paolo Guiotto: And this gives a contradiction, because if this is true, this is true also at times t equals 0. So y equals 0, Y of 0 should be 0, but that's 1, and that's the contradiction.
54:55:530Paolo Guiotto: Okay? So now you understand why it's important to determine constant solutions, because they are boundaries. We cannot cross these borders, because it's like if…
55:06:810Paolo Guiotto: If we cross two different solutions. So, now this proves that that claim is true, so we can go back to our discussion. Now we know that Y second is 2, y, Y prime, we know that Y prime is positive, we now know that also Y is positive. Conclusion, Y second is positive.
55:24:320Paolo Guiotto: Okay?
55:27:850Paolo Guiotto: So Y second is positive, and we deduced that Y is… convex.
55:35:410Paolo Guiotto: So…
55:37:260Paolo Guiotto: Have we solved the equation? Have we found the second derivative explicitly? No. Okay? So you see that we are drawing the same kind of information without knowing what is exactly that function. That's the idea behind this method.
55:55:150Paolo Guiotto: Okay, let's take a break now, and then we continue exploring this problem.
56:07:860Paolo Guiotto: Okay.
56:18:250Paolo Guiotto: Okay, so, what do we know so far? Well, let's redo the figure with the new informations.
56:26:220Paolo Guiotto: So, the new figure is the following.
56:29:490Paolo Guiotto: This is the plane TY.
56:32:950Paolo Guiotto: Of course, you may think that we are doing a lot of work when we can solve directly the equation, it's easy. Yes, I agree, but that's because we can solve the equation, and what if we cannot solve the equation? Then this work makes the difference.
56:47:260Paolo Guiotto: But let's check… I want to show you what happens if we do this job on an example where we can… we could explicitly find a solution
56:59:880Paolo Guiotto: Relatively easy.
57:01:980Paolo Guiotto: So what we know is that the solution is increasing and now convex, so we see a line like that.
57:11:340Paolo Guiotto: With the concavity pointing upward.
57:15:440Paolo Guiotto: So we cannot see something like this, for example, okay?
57:19:720Paolo Guiotto: Or a change-off, and it is always above this, red line, so the x-axis.
57:26:610Paolo Guiotto: Now, a first important question is, what can be said about,
57:31:450Paolo Guiotto: the, let's say, first time of life of the solution, this time alpha. Is it possible that alpha is there, is finite, in particular, or the solution can survive in the past, up to minus infinity?
57:49:130Paolo Guiotto: Of course, we already know that the solution in this case, this was the solution.
57:56:220Paolo Guiotto: And the solutions… solutions are defined until minus infinity for this thing. We already know the answer, but…
58:03:530Paolo Guiotto: Let's see if we can get the same answer here without solving the equation again. Now, here there is a very important… we have to open a parenthesis, a very important property of the solution, the general property.
58:19:590Paolo Guiotto: Which is, this, this, factor that we can call it the,
58:28:800Paolo Guiotto: Well, let's say the departure…
58:35:190Paolo Guiotto: from… Oh, departure.
58:38:840Paolo Guiotto: The exit… from Compact.
58:51:300Paolo Guiotto: sense.
58:53:560Paolo Guiotto: So, there is this remarkable property that solutions have. So, assume… assume… F equals FTY.
59:06:180Paolo Guiotto: verifies… Existence and uniqueness. So, existence and… Uniqueness.
59:18:610Paolo Guiotto: assumptions.
59:24:730Paolo Guiotto: on D.
59:27:20Paolo Guiotto: And let's help the intuition with the figure. As usual, this is the plane, TY.
59:34:70Paolo Guiotto: And this is our domain div.
59:38:170Paolo Guiotto: Then we have a solution.
59:41:10Paolo Guiotto: of some Cauchy probability 0YC.
59:45:30Paolo Guiotto: Now, a remarkable property that we can say intuitively and roughly, let's say, that is maybe easy to understand, but then it is difficult to formalize, is the following property. A solution can never die inside the domain.
00:03:190Paolo Guiotto: So it cannot be like that. Why?
00:06:60Paolo Guiotto: Because, suppose that the solution dies here.
00:11:70Paolo Guiotto: this is the last time of life of the solution. Well, we may think that if we set a Cauchy problem just at that final point.
00:21:450Paolo Guiotto: The theorem says there is a solution of the Buchy problem at that point, so there is another solution that goes a little bit beyond the previous one.
00:32:440Paolo Guiotto: So this would mean that if we glue the black and the red, we would have a unique solution that is alive for a longer time with respect to the black solution. So we could extend the time life of the solution, and this is… this is forbidden.
00:50:390Paolo Guiotto: From this statement, because this statement says that any other solution
00:54:670Paolo Guiotto: cannot be defined more than that interval alpha-beta. So, a solution can never be extended in its lifetime.
01:04:360Paolo Guiotto: So basically, the idea is that a solution cannot end its life inside D. So, in other words, it means that the solution will have to reach the frontier.
01:16:40Paolo Guiotto: of D.
01:17:480Paolo Guiotto: And then it dies, because it cannot go outside.
01:20:660Paolo Guiotto: But to write this formally, it's quite difficult. So the property that it's easier to be stated is the following.
01:27:550Paolo Guiotto: Whenever you take a set K,
01:31:710Paolo Guiotto: Which is compact, so it means closed and bounded.
01:36:850Paolo Guiotto: inside D, while the solution must leave that compact sooner or later, in future and in past. So this means that
01:46:840Paolo Guiotto: if this is a sort of box, where you want to confine a solution, well, sooner or later, the solution will leave that box. So, this is the property. Then.
02:01:690Paolo Guiotto: for every K, Compact.
02:06:670Paolo Guiotto: So, closed the… and bounded.
02:13:370Paolo Guiotto: contained, indeed.
02:17:130Paolo Guiotto: D.
02:18:190Paolo Guiotto: I write in words, then I formalize.
02:21:570Paolo Guiotto: the solution… mask.
02:26:930Paolo Guiotto: Leave.
02:29:420Paolo Guiotto: P.
02:32:480Paolo Guiotto: In.
02:33:930Paolo Guiotto: The pasta.
02:36:670Paolo Guiotto: And in the future.
02:40:100Paolo Guiotto: What does it mean? It means that there exists two times that are the lining times, let's say tau 1, tau 2,
02:51:50Paolo Guiotto: Such that this solution… so what does it mean that the solution leaves the compact? It means that the point on the graph, this point is the point which coordinates TYT,
03:05:390Paolo Guiotto: So the point TYT, is no more in the green box, so it's not in K.
03:13:630Paolo Guiotto: For every time before time tau1, which is the exit time in the pasta.
03:23:790Paolo Guiotto: And for every time after tau 2,
03:28:310Paolo Guiotto: Which is the exit time in the future.
03:33:360Paolo Guiotto: Okay? So, you cannot trap the solution inside the domain, because the solution will still leave that trap, provided the trap is a closed and bounded subset of the domain.
03:49:680Paolo Guiotto: Now, let's see how this is important in this case here.
03:56:00Paolo Guiotto: Because… Let's look at the situation. The situation is, this is the plane TY.
04:04:230Paolo Guiotto: Remind that our domain D is the full plane R2.
04:08:620Paolo Guiotto: Okay?
04:09:940Paolo Guiotto: This is the constant solution.
04:13:220Paolo Guiotto: And this is the, our solution.
04:19:820Paolo Guiotto: Now, I want to convince you that you will see that through this argument, we can prove that alpha must be minus infinity. Let's see why.
04:29:860Paolo Guiotto: Because if alpha were fined, as in the figure.
04:35:390Paolo Guiotto: I could build a box that contains the solution and never leaves this box in the past.
04:43:670Paolo Guiotto: Look at this box.
04:46:630Paolo Guiotto: Example, this one.
04:51:850Paolo Guiotto: You see that the solution cannot leave the box in the past, because to leave the box in the past, it must be defined sometime before alpha, but alpha is by definition the first time when the solution is defined. So, in this case.
05:08:250Paolo Guiotto: If alpha is not minus infinity, so like this.
05:13:780Paolo Guiotto: Then, if I take K, this box, so as you can see, it's just a… it's a rectangle. For the horizontal variable, I have alpha 0, and for the vertical variable, I have from 0 to… this is 1.
05:32:80Paolo Guiotto: Well, clearly the solution leaves the box in the future, but the theorem says the solution must leave the box both in future and in past.
05:43:550Paolo Guiotto: If one of the two exits is not verified, you get a contradiction with this property. So here.
05:51:690Paolo Guiotto: Then, huh?
05:54:250Paolo Guiotto: The solution… Y cannot… leave.
06:03:250Paolo Guiotto: Okay? In. Past.
06:07:460Paolo Guiotto: Because to leave this box impede, to leave.
06:15:130Paolo Guiotto: K. N. D.
06:18:60Paolo Guiotto: passed.
06:21:20Paolo Guiotto: We need… that, we have values, Y of E, defined… for some T,
06:32:920Paolo Guiotto: Which is strictly less than alpha, but that's in contradiction, but…
06:40:320Paolo Guiotto: Our solution is, by definition, defined on alpha-beta, so at the right
06:46:00Paolo Guiotto: of alpha. Alpha, beta, the alpha, and omega of the solution, okay? So, you could not have been alive the day before you were… you were born, okay? That's what it's saying.
07:00:320Paolo Guiotto: So if alpha is, in this case, you see that the box is a closed and bounded set, is contained into the
07:10:130Paolo Guiotto: And that's perfectly fitting the conditions of this. It's a compact container indeed. The solution must live, but this solution lives in the future at time t positive. It's outside, but it should live also in the past, and this is impossible, because we
07:27:290Paolo Guiotto: We have all possible times in that box.
07:30:560Paolo Guiotto: Of course, this argument doesn't work if alpha is minus infinity. Why? Because alpha0 would be…
07:37:620Paolo Guiotto: unbounded in that case, so that box wouldn't be compact, and this wouldn't fit the condition of this theorem, okay? So I can't do this algorithm just because I'm assuming alpha, not infinity. Fine.
07:52:460Paolo Guiotto: So, the conclusion is that this alpha must be
07:56:790Paolo Guiotto: equal to minus infinity. We will see that the same argument does not apply on the right, okay?
08:06:160Paolo Guiotto: We'll see later, but let's say on alpha. Now we know that alpha is minus infinity, so our solution survives in the past until minus infinity. So let's update our figure.
08:19:390Paolo Guiotto: So we still have our constant solution.
08:22:660Paolo Guiotto: the solution of the Cauchy problem we are discussing. Let's look in the past. Well, it is decreasing, because the solution is increasing, so if I go back in the past, it's decreasing. It's concave, it's convex, sorry.
08:39:729Paolo Guiotto: We know that it survives up to minus infinity, and now I want to know what happens at minus infinity.
08:46:910Paolo Guiotto: We know that it goes down.
08:48:840Paolo Guiotto: But, it goes down up to zero, or maybe it goes down to some positive value.
08:55:529Paolo Guiotto: So I want to compute the limit at minus infinity, because the solution
08:59:689Paolo Guiotto: is likely to have an asymptote at minus infinity, and no reason to assign total. A finite limit, for sure. So, what can be said?
09:09:390Paolo Guiotto: So, what… about… the limit for T going to minus infinity of YT.
09:22:220Paolo Guiotto: Again, I will do this without knowing the exact expression for Y of T. Of course, if I have Y of T,
09:30:760Paolo Guiotto: You see, this is the Y of T, exactly… the limit is trivial. You send t to infinity to minus infinity, you get 1 over infinity 0. We know that the limit must be 0, but that's because we have the formula. But what if we do not have the formula?
09:46:689Paolo Guiotto: Well, we can say, in this case, we can say something, because, first of all.
09:55:170Paolo Guiotto: The first thing can be said that
09:57:820Paolo Guiotto: notice that I do not have the formula, so it's not immediately evident that even the limit exists. But since the function is monotonic, so it's increasing.
10:11:460Paolo Guiotto: For sure, there exists a limit.
10:15:10Paolo Guiotto: when T goes to minus infinity of YT.
10:19:350Paolo Guiotto: What can be said about this?
10:25:520Paolo Guiotto: Yes?
10:30:360Paolo Guiotto: Okay, that's exactly what we can say. It is between 0 and 1. It cannot be negative, because to be negative, you should cross the red line. We know the solution is positive. It is less than 1, because this function at 0 is 1, then it decreases going to the left.
10:46:240Paolo Guiotto: Yeah, that's not yet the value, but we know that it is some value between 0 and 1. Let's call it L. And now, let's… I show you how to determine exactly that L.
11:01:900Paolo Guiotto: No, no, in fact, it will be zero. We already know that the answer is zero. But I cannot write greater than zero. Yes, Y of T is strictly positive, because it cannot cross the red solution. But when you do the limit, the limit could be 0.
11:20:830Paolo Guiotto: Why not?
11:22:130Paolo Guiotto: So you can do the limit of positive things, and you get zero.
11:27:560Paolo Guiotto: 1 over T, when t goes to infinity is 0, but the function is positive.
11:31:890Paolo Guiotto: You see? So…
11:36:210Paolo Guiotto: And this… yeah, I cannot say. It is why at point minus infinity, it does not make any sense, because there is no t equal minus infinity. It's just an agreement. We are taking the lead. Now, how do we get this?
11:50:720Paolo Guiotto: Again, we have the unique, the unique thing we can use is the equation, so let's go back to the equation. We know.
12:00:710Paolo Guiotto: That Y prime is equal Y square.
12:04:760Paolo Guiotto: And we do the same thing we have done with the derivative to compute YSEC on the… so the unique difference is that here we need to do the limit. So let's do the limit in the equation, let's see what happens. We know that Y of T goes to L, so what will happen to Y of T squared?
12:22:900Paolo Guiotto: it will go to L square.
12:25:30Paolo Guiotto: And, by the way, this is a finite number. Since L is between 0 and 1, this would be, again, between 0 and 1.
12:34:510Paolo Guiotto: But this says that Y prime of T
12:38:610Paolo Guiotto: which is Y squared, goes to this number, L squared, that is between 0 and 1.
12:45:610Paolo Guiotto: And this is interesting, because it is saying that the slope of this function at minus infinity is going to that value L square.
12:56:90Paolo Guiotto: Between 0 and 1.
12:58:300Paolo Guiotto: Now, think with the intuition. I don't want to formalize this thing, but let's think with the intuition. Now, this is the angular coefficient of the solution, okay, of a function.
13:11:110Paolo Guiotto: Imagine that you go to what? What does it mean?
13:16:40Paolo Guiotto: Y prime at minus infinity is 1.
13:20:750Paolo Guiotto: No! If it is one, the derivative.
13:27:850Paolo Guiotto: Y prime equals 1 means that the slope, yeah, it's 45 degrees, so I would see a function which is,
13:37:850Paolo Guiotto: If, for example, it goes to one, if,
13:41:960Paolo Guiotto: elsewhere is 1, let's say, I would have that Y prime t goes to 1. But this means that it should be a function that at infinity has a slope at 45 degrees, like this one.
13:55:520Paolo Guiotto: And this is in…
14:01:220Paolo Guiotto: Yes, you see, it should go down, no?
14:04:830Paolo Guiotto: But this function, instead, cannot have such a slope, because it's bounded below by that axis that can never be crossed, so I cannot have a negative slope. Does it change if L square is any positive value?
14:22:450Paolo Guiotto: since L square, if I say that this is positive.
14:28:930Paolo Guiotto: This loop goes through elsewhere, okay? It's a positive quantity. Doesn't matter how much positive is, maybe it's very little, but…
14:37:600Paolo Guiotto: If it is… imagine that elsewhere is this globe. It means that your function should have this slope at minus infinity.
14:47:410Paolo Guiotto: Maybe very slowly, but it'll go down to negative infinity. So, you see that it cannot be any value positive that L square.
14:56:790Paolo Guiotto: So, this is impossible.
15:01:530Paolo Guiotto: We'd done… Y of T… going to L. Fine.
15:10:410Paolo Guiotto: Because this would say that the function would go To mine or something.
15:18:620Paolo Guiotto: Okay? Instead, the function is going to assign its value, and therefore means that there is a unit possibility. What is the possibility?
15:30:820Paolo Guiotto: L square equals 0.
15:33:140Paolo Guiotto: And that means L equals 0. You see that we found the value of the limit without knowing the function itself.
15:41:40Paolo Guiotto: So we got the same information we could have by computing explicitly the limit. So with this new update, the plot of the solution becomes more and more precise. But now we know that the solution is made like that.
15:57:330Paolo Guiotto: It goes down to zero.
16:00:230Paolo Guiotto: And now the question is, what happens on this side?
16:04:690Paolo Guiotto: of this story, okay?
16:07:490Paolo Guiotto: So… Now the question is, what… about…
16:15:650Paolo Guiotto: In that time in the future, so beta.
16:19:880Paolo Guiotto: So beta is the last moment in future where the solution is alive.
16:27:380Paolo Guiotto: Now… In this case, I…
16:34:90Paolo Guiotto: I show you that we cannot use the argument of the compact sets to get any contradiction, because, you see, if bettas in the figure is finite.
16:46:820Paolo Guiotto: What can be said? I could say, well, the solution cannot leave this box.
16:52:410Paolo Guiotto: Apparently, I have a compact box, okay, that contains the solution forever in the future. It lives in the past… yeah, that's…
17:05:200Paolo Guiotto: That's acceptable, but that's not acceptable that it doesn't live in the future.
17:10:930Paolo Guiotto: But think about how this box is made. This box would be zero.
17:15:330Paolo Guiotto: B… Times… well, this is 1, up to what here?
17:22:200Paolo Guiotto: What is the value I have here?
17:26:20Paolo Guiotto: Well, it should be the biggest possible value of the green line. The difference before, with the previous case, was that I know that the function is, in any case, bounded below by zero, so it cannot
17:38:460Paolo Guiotto: be less than 0, no? So the values are between 0, 1, at least that left. But at right, I should say, what is this value here? This value… capital L? Well, capital L is the limit value, is the limit when t goes to beta.
17:57:700Paolo Guiotto: of the solution.
18:00:280Paolo Guiotto: And who knows if this value is finite?
18:05:630Paolo Guiotto: Well, we can say, if the value L is finite.
18:10:170Paolo Guiotto: this… this yields a contradiction. So… If beta, he's fined.
18:18:450Paolo Guiotto: And… Also, the limit value, L,
18:24:370Paolo Guiotto: limit when t goes to beta of the solution.
18:29:170Paolo Guiotto: is fine.
18:30:880Paolo Guiotto: Then… We have a contradiction.
18:40:450Paolo Guiotto: Because the solution does not leave in the future that blue box.
18:45:730Paolo Guiotto: solution does… knots.
18:50:750Paolo Guiotto: Lever.
18:53:270Paolo Guiotto: K, which is explicitly zero beta, times 1L, in future.
19:06:350Paolo Guiotto: And this is impossible. So it means that either beta is not finite.
19:13:410Paolo Guiotto: or L is infinite, but they cannot be both finite.
19:18:540Paolo Guiotto: So, from this, I can deduce…
19:21:820Paolo Guiotto: Beta is equal to plus infinity, or…
19:26:460Paolo Guiotto: L is equal to plus infinity. At least one of them must be.
19:31:150Paolo Guiotto: So this is what I can get most from this argument. So, as you can see, I cannot say beta is finite or not, because it could be L equal plus infinity.
19:43:290Paolo Guiotto: Okay? So this is… this is a problem, because it seems that we cannot go beyond this. Well, here there is a trick that allows to say which one of these two is correct. Okay.
20:02:740Paolo Guiotto: And it seems as to solve the equation by separation of variables, and in fact, it is based on the same idea, but we don't solve the equation. Look, we know that Y prime is equal to Y squared.
20:16:630Paolo Guiotto: Since our Y here is greater or equal than 1, we can divide by this right-hand side, and this equation is equivalent to Y prime divided Y squared equal 1.
20:28:940Paolo Guiotto: Now, we integrate this identity from the initial time 0 to the final time beta. So, we have integral from 0 to beta of Y prime over Y squared.
20:42:240Paolo Guiotto: equal integral 0 to beta of 1. The point is that the integral of 0 to beta of 1 is
20:49:760Paolo Guiotto: is beta. So I got the formula of beta. It says that beta is equal to the integral from 0 to beta of y' over Y squared.
21:01:200Paolo Guiotto: Now, this one can be somehow computed. I don't compute with Y of T. I won't use Y of T, okay, explicitly.
21:10:390Paolo Guiotto: This is the derivative of… is the derivative with respect to T,
21:18:820Paolo Guiotto: over minus 1 over Y, no? Because if you do the derivative, you get minus Y to minus 1 is minus Y to minus 2 times Y prime.
21:29:470Paolo Guiotto: So, this is integral from 0 to beta of the derivative of minus 1 over Y prime.
21:37:650Paolo Guiotto: You can, integrate, fundamental formula of integral calculus, final value minus initial value, minus 1 over Y of beta.
21:49:770Paolo Guiotto: minus, minus, plus 1 over Y of 0, which is known, it is 1.
21:56:210Paolo Guiotto: And from this, we see that,
21:59:150Paolo Guiotto: If y of beta is plus infinity.
22:03:150Paolo Guiotto: You see that beta is finite, because that fraction is, is,
22:12:370Paolo Guiotto: is, is… becomes zero, okay? So… If…
22:17:920Paolo Guiotto: Y of beta, which is our L,
22:21:320Paolo Guiotto: is equal to plus infinity, then we get that beta, this quantity is 0, is equal to 1 over Y0 is equal to 1 over 1.
22:32:990Paolo Guiotto: So it is financed.
22:35:930Paolo Guiotto: And therefore, we see that beta is finite.
22:41:410Paolo Guiotto: So, at the end, we arrive, but this is, let's say, a particular
22:46:30Paolo Guiotto: trigger to say that beta is fine.
22:49:00Paolo Guiotto: So we can now, finally, do the plot of the solution. Plot…
22:57:700Paolo Guiotto: Why? So, as you can see, I never computed the dissolution, and what I can say is that my solution
23:08:10Paolo Guiotto: is… well, let's still plot the constant, is alive forever, going down to 0 at minus infinity. This is 1.
23:18:840Paolo Guiotto: And I know that there will be an explosion time, okay? So there should be something like that. What I don't know exactly is what is that explosion time, and only that it is finite from this argument.
23:33:390Paolo Guiotto: Well, of course, this is more precise, because it gives the formula, it tells you what is the final time, you read there, T0 plus 1 over Y0.
23:44:500Paolo Guiotto: So in this case, if you put T0 equals 0, Y0 equals 1, you get 1. So that bit equals 1, it's exactly the answer, but I cannot know in general.
23:54:850Paolo Guiotto: But at the end, we can say that we produce the same output. So, if I don't know
24:00:840Paolo Guiotto: how to solve the equation, I have at least a qualitative idea on how the solution is made.
24:07:350Paolo Guiotto: Now, the techniques you have seen in this example are basically the techniques you can see in other examples. So, I will start an exercise.
24:18:390Paolo Guiotto: Which is exercise.
24:22:230Paolo Guiotto: 4, 10, 1… After you do two, three exercises, you see that most of the ideas are repeating.
24:30:600Paolo Guiotto: The unique point is that you have to adapt time to time, because there is nothing standard here.
24:36:240Paolo Guiotto: So… consider… the… equation.
24:42:710Paolo Guiotto: Here, it starts with an equation. Y prime equals y to y minus 1.
24:51:180Paolo Guiotto: Question 1… Discuss… well, I write shortly, discuss… local… Existence and Uniqueness.
25:08:70Paolo Guiotto: Number two, determine Well, here it is written, stationary…
25:18:470Paolo Guiotto: solutions.
25:20:270Paolo Guiotto: What are the stationary solutions and the constant solutions? Stationary constants are synonymous. It's the same of constant… solutions.
25:34:280Paolo Guiotto: Because the idea is that if you think this is a system where the variable is time, a constant solution is a state where nothing happens forever. So, states there, stationary solution.
25:47:90Paolo Guiotto: And, regions.
25:56:530Paolo Guiotto: off.
25:59:600Paolo Guiotto: equation domain.
26:05:30Paolo Guiotto: where… solutions… Increasing or decreasing.
26:16:90Paolo Guiotto: Number 3… We have YB.
26:20:700Paolo Guiotto: the… solution of her sheet problem.
26:26:690Paolo Guiotto: with the initial condition Y0 equal a generic Y naught positive.
26:35:630Paolo Guiotto: determined.
26:37:930Paolo Guiotto: monotonicity, And… concavity.
26:46:780Paolo Guiotto: of Y.
26:53:740Paolo Guiotto: Determined.
26:57:190Paolo Guiotto: Well, the life span, so if alpha is equal to minus infinity, or alpha is finite.
27:09:460Paolo Guiotto: The same for beta. Beta is plus infinity, or beta finite.
27:18:840Paolo Guiotto: and limits.
27:23:710Paolo Guiotto: of… Why?
27:25:840Paolo Guiotto: Oz.
27:27:680Paolo Guiotto: at… T equals alpha and beta.
27:32:830Paolo Guiotto: And then, number 4, graph.
27:37:780Paolo Guiotto: Off.
27:39:120Paolo Guiotto: What?
27:40:410Paolo Guiotto: So we have a few minutes, so let's see what we can do here. Then you try to continue, we'll see, we'll finish on Monday.
27:48:900Paolo Guiotto: So, number one, we start discussing existence and uniqueness. The equation is Y prime equal y times e to Y minus 1. There you see the function F. So, let… let's start with F.
28:04:250Paolo Guiotto: F of TY, is the right-hand side of the equation is just Y times C to Y minus y.
28:16:430Paolo Guiotto: It is defined on Domain D,
28:22:860Paolo Guiotto: As you can see, there is no condition for T, and also no condition for Y, because that quantity is always defined, so again, we have R2 as domain.
28:32:330Paolo Guiotto: It is evident that, clearly, F is a continuous function on the domain G, so on F2.
28:43:80Paolo Guiotto: and also the DYF. If you want, we compute DYF
28:47:530Paolo Guiotto: is, we don't need this explicitly, but we can later. So it is E toy minus 1 plus Y times e toy, so it's clearly a continuous function on R2F. And there are four, local
29:05:350Paolo Guiotto: local.
29:06:930Paolo Guiotto: Existence and uniqueness holds on D equal R2.
29:17:480Paolo Guiotto: Now, this is,
29:19:90Paolo Guiotto: the discussion about local existence significance. We can apply the theorem. So let's start drawing a figure. The domain is, again, the full Cartesian plane R2, so there is nothing particular to
29:34:330Paolo Guiotto: Draw for the moment here.
29:36:890Paolo Guiotto: Question 2.
29:38:650Paolo Guiotto: It says, determine the stationary solutions. These are the spatial solutions that, as you have seen in the previous example, can be very useful. So, why?
29:50:560Paolo Guiotto: constantly equal to C is a solution.
29:54:690Paolo Guiotto: If and only if we plug into the equation.
29:57:890Paolo Guiotto: The question is Y prime equals something. The Y prime for constant solutions will always be equal to zero, no? So, 0, that's the Y prime, equal… then the right-hand side is y to y minus 1. Y is constant equal to C, so C
30:15:230Paolo Guiotto: E to C minus 1. That's the algebraic equation for C. When this is 0, if and only if we have two possibilities, either C is equal to zero, or E to C minus 1 is equal to 0.
30:29:940Paolo Guiotto: Which means, E to C.
30:32:940Paolo Guiotto: Equal 1, and when this is possible.
30:36:260Paolo Guiotto: C plus Z. So, as you can see, there is a unique constant solution, so… unique.
30:47:810Paolo Guiotto: Constant.
30:49:830Paolo Guiotto: solution, Y constant equal to 0. So that's the constant solution.
30:57:980Paolo Guiotto: Then, the question asks also to determine the regions of the domain where solutions are increasing decreasing.
31:05:530Paolo Guiotto: So now we notice that a solution Y is increasing if and only if, as we say, the Y prime is positive. But Y prime, because of the equation, is y toy minus 1.
31:20:340Paolo Guiotto: So we need that this quantity be positive.
31:24:260Paolo Guiotto: Now, let's rewrite here. YE2Y minus 1. This is, even if you don't see T, this is an inequality for TY. In this case, there is no T, so it is easier. Now, that product is positive.
31:40:600Paolo Guiotto: Either, if both factors are positive, or both factors are negative. So I evenly…
31:47:300Paolo Guiotto: Y is positive, and e to y minus 1 is positive, or together.
31:54:460Paolo Guiotto: Y is negative, and E toy minus 1 is also negative.
32:01:50Paolo Guiotto: If you think about when y is positive, the second equation is… inequality is e toy greater than or equal than 1.
32:11:300Paolo Guiotto: Which is exactly when Y is greater than or equal than 0. So, this is redundant. And similarly for the other case, Y less or equal than zero, or E2Y less or equal than 1, which is, again, Y less or equal than 0.
32:25:350Paolo Guiotto: So what is the conclusion?
32:27:40Paolo Guiotto: That quantity is positive if and if Y is positive or Y is negative.
32:32:190Paolo Guiotto: So… For every one.
32:34:860Paolo Guiotto: And actually, a bit more precise, for every PY, because even if you don't see T, there is T, okay, for every TY.
32:44:410Paolo Guiotto: So what does it mean? That the region where the solutions are increasing is the interior plane? So wherever is this solution, it is here, it will be increasing. It is here, it will be increasing.
32:58:560Paolo Guiotto: It is here, it will be increasing. It is here, increasing. So, as you can see, all solutions are increasing.
33:08:430Paolo Guiotto: Okay.
33:10:120Paolo Guiotto: Now there will be the number 3, which is quite longer, so let's see.
33:15:570Paolo Guiotto: However.
33:17:260Paolo Guiotto: A point important is that you have tools to understand what is going on before you discover, really. So, now, why is the solution of the Cauchy problem Y0 equals y naught? What is it?
33:30:600Paolo Guiotto: It is, when not is positive, it is about here.
33:35:650Paolo Guiotto: So now, if I look at this figure, I immediately should understand that the solution is increasing, so I start drawing a solution like that. I don't know the concavity, but it is increasing. So if I go back in the past, the solution goes down.
33:51:270Paolo Guiotto: Up to where? That, well, it's similar to the previous exercise. You cannot die inside here, so probably you expect that you'll go down up to minus infinity, and probably you will go down to zero.
34:05:580Paolo Guiotto: For a similar argument. About this side, I don't know. It should be discovered if the solution blows up, if the solution goes to some limit.
34:15:910Paolo Guiotto: Of course, if we know that the solution is convex, this case cannot be. So the study of convexity will tell if this is possible or not, okay?
34:26:950Paolo Guiotto: So, however, this is, so far, what I can expect. Moving forward and repeating the same kind of calculations we have seen in the previous exercise, you should be able to, to try to finish this exercise. So.
34:40:530Paolo Guiotto: complete this, and on Monday, we will see together the solution, doing other examples, okay?
34:49:260Paolo Guiotto: Yeah, on Tuesday, sorry. Sorry. On Tuesday.
34:53:830Paolo Guiotto: Have a nice, day, weekend, whatever.
34:58:200Paolo Guiotto: Thank you.