 Good morning to everybody. This is a course on ordinary differential equations, ordinary differential equations popularly known as ODE. So, we will be using this ODE course to represent ordinary differential equations. I am A. K. Nandakumaran from Department of Mathematics Indian Institute of Science, Bangalore. In this course we will be basically covering a standard university syllabus. So, all the material covered in most of the Indian universities will be covered in this course, but in addition to the normal syllabus we also plan to introduce certain notions and especially some topics like face plane analysis, face portraits, what is called as the qualitative analysis of differential equations. So, it is a little more material than what is usually introduced in the university. Thus this course is going to be very useful not only for the students, especially the MSc students of Indian universities and institutes. It will be really beneficial to the faculty or the teachers who teach this course because they get a little more idea than what is normally covered in our place. And we want to see this course in a entirely different perspective. In this course, we cover approximately in 40 to 45 lectures and three of us are involved in teaching this course, P. S. Dutty from TIF, Tata Institute of Fundamental Research, Bangalore. It is basically called TIFR CAM, here for applied mathematics and Raju K. George from IIST in the Institute of Space Science and Technology and of course, myself. In the first lecture, we would like to plan to give an overall view of the entire course and at the end of it, we will explain to the end of this first lecture, we will explain the material covered in this course. Before that, we will give a various issues, some aspects of this course, why we want to this course and what are the prerequisites required to learn this course, we will be introducing all these things. So, this lecture, the first lecture going to be very general and students will get and the teachers will get an overall idea of this entire course and that will help them immediately to go to a particular module, if they are familiar with certain other modules. So, they do not have to, if they are familiar with certain aspects of differential equation, say all your existence theory, linear equations, everything if you are familiar, you do not have to spend time and you can straight away go into the analysis of linear and non-linear system, phase line, phase portrait, which probably some of you may not have seen it. On the other hand, if it is a student for the first time and if they want to learn differential equations, they can start including the basics. So, what are the, so let me begin with the prerequisites required for this course, prerequisites. We expect the, both the students and the teachers are familiar with a first course on linear algebra, we will elaborate little bit there and then the second thing, what they need to know about the some notions from analysis, some notions from analysis, like convergence, especially the uniform convergence, like notions like Lipschitz continuity, something like what are called Azela Ascoli theorem and some theorems like Banach fixed point theorem, fixed point theorem and linear algebra, the notions like basis, independence and you also need to know the concepts like linear operators, matrix representation, representation, the concepts like Eigen values and things like that, Eigen values, etcetera, you want to know that. So, this is what the prerequisites we expect from this course to fully understand, but to make this course a self sustained course, we will be spending few hours, probably 5 to 6 hours in recalling these notions, so that those who are not fully comfortable with these things get to know what they exactly want to know, but we definitely recall the entire material, basic material we need to know, we may not be able to prove each and everything, because it is basically one or two courses, additional courses which you should be familiar with, just to tell you that what you should be familiar with and you can get into that, if you are not understood fully from this course about the prerequisites, you can go back and study that material to understand this course. We want to say, before proceeding further, we want, I would like to explain to you why we have planned to introduce this course to you, why this course, we want to spend few minutes, let me tell you why this course. As you know very well, differential equations lies at the heart of analysis, this is the one factor, whether it is ordinary differential equations or it is a partial differential equations, we feel that the differential equations really lies at the heart of analysis. And if you look at the history of the development of science and mathematics in particular and more precisely the analysis, the root cause of development of all these notions lies at the differential equation, because differential equations models the physical systems and to understand the physical systems, you have to learn the differential equations and to develop a systematic theory which will tell more about it soon and analysis and other mathematics had to be developed and that is what happened in the history of here. For example, if you see the development of, is starting with the integral calculus and the basic notions of analysis and then even for the initial developments of complex analysis, these are all happened in probably the 19th century and then the 20th century, the very powerful functional, the development of functional analysis, then operator theory related to that and this really exploited the development. And if you look at really all these things, you can really see that differential equations is the root cause for developing all these things. But the one of the unfortunate thing is that the analysis themselves have forgotten this fact, the differential equations are the root cause for these things and it is not especially when you are developing the, when you develop the a curriculum or a syllabus for differential equations, this fact is not taken into account at all and in the process the beauty of differential equations is lost and especially the interplay between differential equations and with other subjects like analysis, even algebraic topology, differential geometry and all the beauty is lost. So, if you want to understand differential equations in a beautiful way and impact on other things, you have to understand a certain interplay between the differential equations and the other subjects. So, we will not be able to do it the entire thing, but we want to address this course or teach this course in our period to see the impact of at least the algebra and geometry. But this is a completely different from what you see in a normal standard university course, where a set of artificially created methods are introduced to the students and most of the students think that differential equations is a course consisting of a set of methods and even the set of methods, they may not be aware how the set of methods are coming up, because if you want to introduce the set of methods, you have to get into the analysis and other aspects, which is not exhibited in our teaching or even in the syllabus it is not exhibited. So, our main motive in this course is to see some aspects of algebra and analysis. As I mentioned that is why this preliminaries you have to prerequisites you have to go through it, but definitely as for example, if you want to see one example to show that in this introduction, look at the eigenvalues of the matrix. It is introduced in a very casual manner, but most of the students do not get an impact of this eigenvalues, but after studying this course, you can see that how the second values are and its Jordan decomposition are used so powerfully in analyzing the linear and non-linear systems and especially understanding the stability analysis phase plane phase portrait, you can see. So, it automatically motivates if you understand the linear algebra, it motivates not only the differential equations, it is also get good enough motivations, why we study analysis the eigenvalues of a matrix and more generally in functional analysis the use of spectral theory and all these developments in the other subjects of course, in fact, help you to understand differential equations. So, it is a give and take policy and which we want to show it to the students and that is why we want to see the course in this one. So, let me continue little more about that one little more about this introduction to differential equations, probably the beginning of differential equations started from the attribute to the development attribute to Newton and Leibniz after the invention of the differential and integral calculus. So, the early work comes from Newton, Leibniz and Bernoulli probably starting from the late 17th century and there are other people like in addition to the Newton, Leibniz basically the main people Newton, Leibniz, Bernoulli all these early early problems. That is part of the century, many Bernoulli there are two Bernoulli and then there are other many other people contributed to something like Euler, Lagrange, Laplace there are many people there are I will not be able to write some all the names like Euler, Lagrange, Laplace, Abel, there are many many people like Poincaré and many other people over the four year of course, you cannot miss that four year Gauss and many other people. And if you look at in the early part of this thing the main issue was to address the physical problems. So, if you have a particular physical problem your model it and the model turns out to be differential equations may be ordinary differential equations it can be partial differential equations or integral equations is anything. So, the issue was to solve that differential equation you also have to understand this is a pre-analysis era the 17th century and early part of it and 18th century or all pre-analysis era where even proper definitions of functions, convergence, continuity or all not available. So, the idea was to obtain the solutions to your differential equations which represents the which represents actually the physical models and derive solutions in whatever way obtains functions in the simple form and you predict your physical phenomena and that is and the only methods in this directions were integrating factors and separation of variables etcetera. But soon in the people like Euler and other mathematicians realized that making an attempt to solve differential equations is futile and the reason is that they were trying to develop even though it is a each differential equations represent different physical phenomena and trying to solve it separately they want trying to develop a systematic theory for the differential equations that was the beginning of the thing and the realization came that making a devising explicit methods to solve differential equations is a futile attempt and that even stands today the differential equations which you can exactly solve are very very limited and most of the practical problems there are no explicit solutions that is where it. So, the especially it is in this situation scenario you have to see the qualitative analysis like existence, uniqueness theory, stability analysis, large time behavior or that you want to do it and that is where the analysis geometry and linear algebra and other mathematics development is important to understand the differential equations. So, really a second phase of differential equations started from the beginning of 19th century where the most of the analysis also started to develop. So, we will see these things in the coming lectures. So, let us begin. So, what is the basic differential equation? If you look at what the most easiest differential equations probably can be coming from d y by d t is equal to f of t you see and then. So, this problem all of you know after the study of integral calculus of this problem is nothing but your integral calculus problem this is just to convey to you that integral calculus problem. So, what you say given a function you want to determine y of t. So, attempting to solve this precisely your fundamental theorem of calculus and other things will come into play and the beginning of equations this is of course, Newton's second law of motion you will have the all of you are familiar with this kind of equations which represents the second law of motion of course, this can be written as a first order system if you put x dot is equal to v then you will have m v dot is equal to minus f of t. So, you see you can view this second order of equations thing. So, here is a one point I want to tell a little more about these things as we go along this introduction and throughout the lectures you will see the importance of these equations. And another way of the thing you view differential equations as a dynamical system and this is the view we would like to pursue throughout this most of the time we will be pursuing this thing dynamical systems where you can view y t is a trajectory of a or a motion of a particle in some space if it is a three dimensional it will be a motion of the particle you can say the motion of the satellite or you can say the motion of a missile or motion of any planet or anything. So, the other thing that way the dynamical system point of view gives a better feeling about your differential equations. And you can see a plenty of example you are going to see different examples different thing throughout to this course you will be seeing it. And what you say again you see that t is your independent variable and y is your dependent variable. So, given time t given independent variable t you will see your solution y t which is a trajectory of this thing. So, let me say just this. So, what is a differential equation basically what is a differential equation what is a differential equation as you see which you are going to see is a formally it is nothing but a relation. So, you will have a independent variable t and you will have an unknown function which a function of y t and you will have its derivatives it depending on what type of problem it can also come d square y by d t square. And so on you can go up to d power n by y by d t power n and you will have a relation connecting this course. So, formally I can define a differential equation is a relation connecting a independent variable in if it is a motion of a trajectory you can think t is a time variable, but it not necessarily the time all the time you will see other types of examples. And y is the unknown variable and the its derivatives in relation connecting with that one and this n is basically called the n is the order of the differential equation order of the differential equation. So, for example, if you have a first order if you want to understand a first order equation first order equation is nothing but a relation connecting t y t and p y by d t even to understand this equation is not easy. So, most of the time you will be seeing a something a simpler form of this equation where you will be having d y by d t is equal to g of t y t. And this is the differential equation which we may most of the time you will be addressing it as I say here this and this are not the same if you want to get you should be able to solve your d y by d t from this relation in terms of t n y and in general that will not be possible. So, but we will restrict to a different because this is a much general category this is a more general category and this one is a particular case of this differential equations. And for a second order equation if you want to see that we will not go further second order you will have a differential equation connecting t y of t and d y by d t you see d y by d t equal to sorry you will have d square y also you will have d square y by d t square is equal to 0. So, this represents a second order equation and we will in addition to that there is a classification called linear differential equation and non-linear differential equation. So, we will have a set of lectures on first order second order linear equations and also an nth order which will be converting to a system of. So, we will have a separate analysis separate study about this equation and then there will be a general study of first order equations as well as the first order system we will do it is a studies and we will explain little more about this thing. So, if you have more than one independent variable instead of just one independent variable one independent variable t and one dependent variable y you have an equation, but on the other hand if you have more than one unknowns it will lead to a system of differential equations. On the other hand if you have more than one independent variable say t s etcetera and it will lead to partial differential equations. So, we will not be doing anything regarding the partial differential equation in this course and that is an entirely something different. So, now we want to explain little bit about what is called a initial value problem because what we are going to do is initial value problem. So, let me explain which is normally called IVP. So, whenever you see IVP it is nothing, but initial value problem. So, let me start with your integral easiest problem called the integral calculus problem y t is equal to f of t. You have to understand that f of t depends only on t. So, when you are given such a relation the integral calculus problem tells you that you want to find given f you want to solve your f of t and this is formally you can write it as all of you would have seen this thing you write this as integral of say I use the word f of s d s or f of t d t does not matter. So, you will write f of t d t and probably a constant. This is very clear because if you take if you have one solution to this problem integral calculus problem which you are familiar at the twelfth level then you can add a constant to that you will get this that will also be a solution. So, the solution definitely is not unique if you have one solution you will have can always add a constant to get another solution. But what you essentially telling in the entire integral calculus problem what the fundamental theorem essentially tells you if you really look at the fundamental theorem carefully it tells you that these are all the solutions which you want to do that. So, if you have one solutions of this form and the integral calculus tells you that if f is a continuous function which will not be covering here, but you can see that such a thing you can be defined the integral can be defined using the concept of area that is another issue and what the finding tells you that all solutions to this problem are given by this one. So, this indicates if you wanted to have a particular solution. So, you can think that as I said y t may be representing a trajectory. So, there will be one trajectory if you have there will be many many trajectories to the solution is possible to that is what if you have some trajectory something like that if you have one trajectory then you can add a constant to that trajectory and you can get many solution. So, if you want to find a particular solution going from here few. So, that is a very natural physical problem you are in a place particular place at time t naught you will be at this point y t naught you will be at this point you will be at this point y t naught at time t naught and then this equation tells you that you know the velocity or all the time this exactly it gives you an f t. So, you are given a position and then you are at the what we call it a initial time this time t naught is called the initial time and then you have the f t is the velocity f t is the velocity at all is the velocity and that is a standard physical problem. If you know the initial position and velocity at all the time and your job is to determine the trajectory. So, this constitutes a reasonable good physical problem d by by d t equal to f of t and then y at t naught your initial position y naught is given to you and this together is called the initial value problem and this can be this really motivate and this can be done for any general thing. So, if you have a general differential equation which we will be addressing in this course d by by d t equal to the f of t it may not only depends on the independent variable t it also depends on the position at that time and y at t naught is y naught is given this is your initial value problem you see you have a very nice way of understanding that. There is a so much difference here because in this case your dynamics what we call it namely the velocity is given to you a priori you do not need to know that that. So, the velocity does not depend where you are it depends only on the time here the problem is that the velocity not only depends on the time and it also depends on the position making it a the in general this is a very non-linear problem it is a highly non-linear problem and that is a and the difficulty what you can anticipate here even such a simpler problem to solve like a simple integral the beautiful theory of integral calculus is developed and it was not an easy job. So, that immediately we can have an anticipation if such a simple looking problem is difficult to solve you can really anticipate a much deeper difficulties in this more general problem. So, you are trying to basically invert a differential operator of this thing and trying to solve that one that is why it is becoming a really a non-trivial theory is required to develop to understand these kind of problems. So, you can really feel the as you go along the difficulties coming here. So, if you have a second order equation say second order equation in a slightly simpler form let me write to here y of t y of t and y prime of t what is the suitable here you may have an options but let me tell you one thing. So, I let me convert this differential equation into a system of differential equation which generally possible. So, I will put y 1 of t is equal to y of t and I put another variable. So, I introduce two dependent variable y 1 which is nothing but y and y 2 is nothing but y 1 prime of t that is equal to y of t. So, with this together if you apply this here what is this one y 1 prime of t is equal to y of t and then y 2 prime of t what is y 2 prime of t is nothing but y 1 double prime of t that is nothing but y double prime of t this is y prime of t. So, y double prime of t that is nothing but f of t y 1 of t and y 2 of t. So, you have a system of two first order equations system of two first order equations first order equations and actually if you look go back and see each first order system. So, if you want to understand the first order system of course, depends on this is y of t. So, this will be y 1 prime of t is nothing but this is y 2. So, this is y 2 of t not y 1 of t. So, if you look at this equation this is a first order equation for y 1 and hence if you look at the previous page what I said that if you have a first order equation you have one initial condition basically because that is a standard and. So, you require one condition for y 1 and another condition initial condition for y 2. So, if you want to have it is a kind of nice problem you need an initial condition for y 1 at t naught and you need an initial condition for y 2 also. So, one of the possible equation so an initial value problem for a second order equation looks like this there may be other possibilities which you may see as the course proceeds. So, you will have y of t y prime of t together with one condition for y at t naught y naught and you need condition for this is for y 1. So, you need a condition for y 2 that is nothing but y prime of t naught. So, it is a y 1. So, this is a standard initial value problem which again you will be studying this equation and especially you will have a little more detail in the study about this equation in the case of linear equation. We will explain when the stage comes to understand the linear equations as this is an introduction we will not get into the detail thing. But as far as second order equations are concerned there is another interesting set of problems called boundary value problems, boundary value problems. Quite often the second order equations may be defined in a just an interval a b there are many many applications especially a set of the equations like coming from stream Liouville systems representing the mechanical problems and vibration problems. So, we are going to spend any way some time on the stream Liouville and other things where you have a second order equations and the conditions are not defined at the initial values, but the conditions are defined at the boundary value. So, the boundary value problem in short form we call it BVP and applications why boundary value problem comes we will also see when we introduce the stream Liouville systems as well as the general boundary value problem and you will see all that one. So, typically a boundary value problem may look like f of t y of t y prime of t and this is for t in a b. So, it is prescribed the interval with you can have very general boundary value problem. So, I can put it something like alpha into y t a plus beta into y prime at a is equal to something like 0 if you want and one more condition something like alpha or some other parameter you can posit when you will have a solution you will see y at b plus beta y prime at b is a general thing. For example, if you take y prime of a equal to 0 or b if you do not a beta equal to 0 alpha equal to 1. So, you have a y a equal to 0 y b equal to 0 that will be the general that will be a particular case of that. So, you will be studying so basically you have an equation prescribed in certain interval and you have the boundary conditions. And why we study just like we have a motivation for the initial value problem it is along the trajectory you will have the motivation when we introduce because I do not want to spend time now on that boundary value problem in the introductory lecture, but you will get to know about it. So, with that we will spend little more time on one more issue. Let us see whether we can complete everything what is something like a solution concept quickly solution concept. There are various things this is a actually a bigger section we will not get into the more complicated concepts of solution concept which are indeed useful in applications, but we want to just tell you something about the concept which I want to introduce. The normal procedure when you have a differential equation which is started 300 years back definitely a differential equation associated with a physical problem you want to get solutions in the form what you like it in the in terms of simple functions or whatever it is. But as you see even for the integral calculus problem d y by d t equal to f of t you already know that there are certain f t you may get a representation y t in terms of integral of f of t d t, but integrating the functions in a very closed form may not be possible you see. So, if you can get a solution y of t if you can write something y of t equal to something this is called a explicit solution you can write explicit solution. As what I am trying to say that writing down your y of t quite often in an explicit this is more or less explicit you may not be able to calculate the integral and write in terms of that, but still this is explicit, but when you go to more general equations of the form d y by d t equal to f of t y t getting an expression in the y of t may not be possible. What you may get is that a relation is possible that you may get a relation connecting t and y possibly t and y even this may not be possible that is why I said solution concepts are different. You can get it here and this is what you most of the time the methods quite often restrict to this one not beyond that. So, connecting t and y and you will see examples as we go along and when we present examples in our study you will see equations where you get relations that connect and this is called what are called implicit form in the implicit form. For example, if you get a solution of the form y square plus t square equal to 1 example if you get suppose your solution represents in this form t square plus y square equal to 1 it is not possible to solve y uniquely which all of you know it. So, you will have a relations connecting t you may have a relation connecting t and y and we call this also a solution to this problem even though you are not able to solve y in terms of t. So, that is a implicit relation. What I want to do here one step further even getting this in the either in the explicit form or in the implicit form may not be possible and that is where your analysis will come into play. That does not mean that this equation the differential equation has no solution if you are unable to get an explicit or implicit form. It is not possible to conclude the exactly that this does not have an explicit form it does not have a solution you may have a solution which you may not be able to represent and that is where the theoretical study of existence and things into come into picture. And the numerical computation of this differential equations comes into play because the engineers and scientists other science people working scientists who do the problem they want solutions. And if you are unable to provide solutions in the explicit or implicit form what you do you cannot stop there you have to tell them that you can do something else and that is one of the major aim of this problem. And in the our university syllabus essentially restricted to these two aspects and our aim of this course is go beyond this. So, I will quickly tell you little more thing. So, what are the things that are the issues we would like to address through the course of course we have to address the issue which you are familiar methods to solve differential equations as we. But we do this thing but our concentration is not just this one how do you get that methods what is ideas behind that methods that is what we will be concentrating. And then the second part I told you you may not have an implicit or explicit relation some of the ideas which you want to do that one existence uniqueness. Why existence is required unless you tell them that there is a solution even to proceed for numerical computation how do you proceed if you do not know anything about the equation and you may try to develop numerical schemes you end up with false results. But madam and if you can tell that you have the existence and uniqueness the you can do the numerical computations and this is where your play in the analysis will come into play. And there is another interesting fact which probably you may not have understood or may not you heard in your university system what are called the continuous dependence one of the very crucial notions together with existence uniqueness and continuous in them. This is for practical purposes because if you formulate if you do a modeling for your differential equation it will be an approximation for example the dynamics f or the initial conditions y naught may not be available to you it may be coming through some data in fact even the explicit form of f may not be available what you may be available is a finite set of data for f using the finite set of data using the approximation schemes you may be able to the approximate f. So, you may not have f and y naught exactly may not be available exactly may not be available what you may not be available what you may get is an approximate value of f bar and y naught bar you see. So, you will be solving the equation you do not have f and y naught, but you are theoretically studying the equation with the using f and y naught, but actually what you may be available may be f bar and y naught bar. So, you are actually may be solving your differential equation with an approximated data, but what is the guarantee that the solution you obtain using the approximate data is an approximation to actual solution you see that one that may not be true you will see examples again in this course even if you use approximated data the solution may. So, you have to need condition you have to get tell them under these conditions you may be able to get an approximate solution and that is what we will be discussing in the continuous data. And then after continuous dependence of course you have numerical thing you are already told you why you need to understand numerical computation because you may not have a solution, but engineers and scientists wants that they want values. And the last and important part which we will be covering in this thing what is called the qualitative analysis qualitative analysis you see. This is also there are multipurpose one of the reasons is that in many situations you may not be interested in actual solution. For example, if y t is the solution your interest is you do not want to know what is y t at each time your interest is the what happens to y t as time progresses you may not be interested in the whole path you may be interested in either limit t tends to 0 you may be interested in limit t tends to infinity or limit t tends to 0 whatever it is of y t and such type of asymptotic behavior or you want to know more about the geometric picture about it. And this you will be seeing in the various things like this is involves in qualitative analysis and you will see the phase plane analysis phase portrait and whether your trajectory is stable whether your trajectory is unstable and all that will be seeing. And our ultimate aim in this course if you are successful is to see some sort of periodic solutions which is the one of the famous theorem called Ponger-Habendixson theorem that we may not we are not sure we will reach that one because giving a proof of Ponger-Habendixson theorem involves much deeper understanding with that let me now quickly in 5 minutes tell you about our module which is already available to you. So, this is our whole course going to be here so we will start with to get you a feeling we will start with some motivation that is what I have given in this lecture I hope it is motivated you to learn differential equations if it not motivated I am sure over the period of time you will get motivated and my colleagues will motivate you more than me as we go along. So, we may spends probably 4 lectures in module 1 you see where after this lecture I may spend around 3 or may be 3 lectures I am not sure we will go along we will give you some real life examples especially like dashboard the thing population growth and some non-linear systems may be a satellite example either I will give it or you will see later and we are giving this examples and we recall this examples as and when required in our course. So, there are many many examples which we cannot give you can give examples of ordinary differential equation in every field of science engineering, but we have chosen some classical example is nothing new most of them are classical examples available in most of the books, but we will do it and we use that examples to do that one that is what in the thing then as I said that next step is our module 2 we will give you the basics which I said here we will do we will give whatever preliminaries which I explained and then to begin with before going to the serious general theory our next module is module 3 in which we will restrict these are the easier equation which you can understand. So, we will spend some time on your first order linearly this is a very special class of differential equation and the interesting thing is that you can convert that problem eventually to an integral calculus problem and many means to motivate you the some of the notions like integrating factors and the concept of exact differential equations. Then immediately you will go to the second order linear differential equation you can see that things are not even for the linear differential second order equation the life is not that easy here you could convert it back, but there is a very nice interesting things here and it is not easy to solve the differential equation on the first order linear differential equation you have a reasonable good theory. Second order also you have the theory, but solving. So, you need some methods of solving also if possible, but we will explain to that in the coming lectures and then we will see the some part of the theory namely the general existence uniqueness theory. So, basically we will give you different methods to prove the existence and uniqueness continuous dependence we will start with the first order equations, but then we will also go to the next thing and there are different methods will be we will also introduce some methods of solving after that we will go to second linear system here the powerful use of linear algebra you will be see it. So, we will represent all the linear algebra and you see some stability theory for the linear systems will be seen here and some periodic solutions and stability will be introduced here and this is a class of boundary where you will see a special we will spend not too much time we can spend more time on one, but we will see some oscillation theorems and example and you will see the Bessel's function Hermit Legendre equations all that interesting example and we will see something a comparison theorems from one differential equation to other equations. And this part our one of the important part of our module which are not really covered not at all covered in the university syllabus is the qualitative analysis already we have done something in the linear system you will see the the various stability analysis like you know stability you will especially the geometry and phase portrait in this 2 D systems and this is our ambition to do something on Pankare Bendix and Hope Wilde. We will also give few lectures we do not spend too much time because it will go beyond our number of lectures we plan for a one semester course is difficult, but we will try to give little bit on the introduction to two point boundary value problems and that is what we are planning to do in this entire course. So, let me complete this thing I have few more minutes in this just I have 2, 2, 3, 2, 4 minutes. So, let me conclude the entire thing with drawing a schematic diagram. So, you have basically you have the physical phenomena physical phenomena you will have the physical phenomena you have the physical laws here and that is what you have started with your or everything the whole science is that right physical phenomena and physical science. These 2 understand the earliest things were observations observations initially that is done if you look at our astronomy everything is observation and that is started with observation in the later part by Galileo and other people who have started experiments is not just experiment you create labs and do the experiments and then of course data collection etcetera want to understand that physical phenomena you have done that. So, here you will have that you see using the mathematics available the physical laws and these things together you do the mathematical modeling you have the mathematical model here you see correct mathematical model. Here is what if you want to understand this one here is the place you have to do the analysis you see this is where you are trying to do if you can have the mathematical model if you can solve if you can explicitly understand everything no problem use the mathematical model using these things in your physical phenomena understand the physics and engineering behind it using that one for that you need the analysis methods to solve it and of course you will have the computations and the aim of the course is in this box we are only here nothing else we are not here we are only here. So, we are restricting to this box and after doing the analysis and mathematics here is where your the development of mathematics coming development of mathematics and once you do the analysis and methods of computations your model may not be very perfect this will also help you this is a two way guy. So, after doing that if the model does not predict you using the analysis properly here use the analysis and mathematics here to improve upon here do further experiments may be further data collection further of thing you improve your mathematical model. So, you will come here further again. So, there is a circle between these things you understand this one on this mathematical model may be this one simpler thing, but you will have other issues like in the mathematical model where you will have control and other things like this is important now because you have a marks mission you see you apply control to grade your trajectory if the trajectories are not taking bath you have to correct it what the ISRO is doing currently. So, I should mention that one. So, that leads to another branch of mathematics entirely called control theory our aim is that not that one, but if you want to spend time yes you can do that with this we will finish this introduction and we will finish this introductory lecture we will start now next with some of the examples real life examples. Thank you.