[0:24]Hello friend, uh, welcome to this lecture. In this uh lecture, we start with the uh decision of uh of a dynamical system. So let us uh discuss what is we are going to discuss. So, uh, first let us define what is uh known as system. So, a system is defined as a collection, set or arrangement of objects, which are related to each other by interactions and produce various outputs in response to different inputs. So if we have, say, some objects, they are collected or they are arranged in a way that uh that uh they are interrelated and they produce different outputs uh corresponding to different inputs, we call that uh kind of arrangement as a system. And a system is called dynamical system if it varies with respect to time T. So for example, uh, you take electromechanical machines, such that motorcar, aircraft, washing machine, all these kind of electromechanical machines are coming in uh under this dynamical system. And biological systems such as human body, and the body of um different uh species, they are all coming in uh dynamical system, and economic structure of countries or regions. So if you take consider any country and consider the economic structure of that country, then it become a system which varies with respect to time T. And we call this structure as a dynamical system structure. And in fact, anything that evolves over time can be thought of as a dynamical system. So any system which evolves over time, we call that as a dynamical system. And if we model that dynamical system in terms of mathematical uh model, then generally, a dynamical system is described by an initial value problem of differential equation. So, in this uh particular lecture, we uh focus on uh concept related to initial value problem, what is initial value problem, what is uh differential equation and try to see certain example of dynamical system at the end. So let us uh start with the history of differential equation. So, the history of differential equation began in the 17th centuries when Newton, Leibniz, and the Bernoullis solved some differential equations of 1st and 2nd order arising from mechanics and geometry. So in the beginning uh all these problems are uh started with the problem of mechanics or problem of geometry, and they try to solve these problems using geometrical tools and all that. And uh in the process, they try to solve some first and second order um differential equation. Now, differential equations are used to express many general law of nature and have many applications in physical, biological, social, economical and other dynamical systems. In particular, the origin of differential equations may be considered as the efforts of Newton to illustrate the motion of particles. These equations may provide many useful information about the system if the equation were formed incorporating the various important factors of the system. So if uh you consider a real world problem, and um then you consider the change in some kind of dependent variable. And that dependent variable if it is a very important factor of the dynamical system, then um uh by looking at that dynamical system, by solving that dynamical system, we are able to uh predict the uh behavior of the dependent variable which which is placed which plays a very vital role in that particular dynamical system. So, um let us uh say that a differential equation is a relation between independent variables. So, most of the time, when you deal with the dynamical system, and when you rewrite in terms of mathematical terms, it turns out to be a differential equation. So, uh most often it is coming out uh to be a ordinary differential equation. So, here we will focus in this lecture and uh in the subsequent lectures also, we'll focus um on dynamical system as a uh differential equation. So here let us start with the what we call as differential equation. So, a differential equation is a relation between independent variable, dependent variables and its first or higher order derivatives. And depending on the number of independent variables, we may classify the differential equations in two parts. First one is ordinary differential equations (ODE), second one is partial differential equations (PDE). So, in ordinary differential equation, there is only one independent variable. Let Y (t) define a function of t on an interval I, where I is some non-trivial interval starting from uh a to B, where B is bigger than A.
[6:05]Now by an ordinary differential equation, we define an equation involving t, Y (t) and its one or more higher derivatives.
[6:18]So any equation which involves the um independent variable, dependent variable, and its derivative, say, at least one derivative or say more than one derivative, which may be first or higher order derivatives, that we call as that equation, we call as differential equation.
[6:40]So here are some example of ordinary differential equation. First one is dy/dt = αy, α > 0. And d2y/dt2 = g, g is some constant. And dy/dt = αy - βy², α, β are some positive constant.
[7:02]And fourth one is md2y/dt² = mg - αdy/dt. So these are some very trivial example of ordinary differential equation. And in fact, all these uh example have some originated from some real world problem. For example, the first problem dy/dt = αy is basically uh represent a very simple example of population dynamics, where this Y (t) represent the population of a given species at time T. And this simply says that when there's no interaction with the the environment, then the population growth or uh can be given by this equation dy/dt = αy. And if you look at the second equation, second equation is basically uh representing the free fall of a particle from a height some height. So it means that when your particle is uh say falling under the influence of gravity and there is no other um say friction or there is no other hindrance, then the motion of the particle can be modelled by this equation d2y/dt² = g.
[8:19]Now, if you look at the equation number three, then equation number three is uh slight modification of uh the equation uh differential equation given in one. Where we have considered one additional term that is uh βy². And we'll see why this βy² is given. So, this is a uh say modification of uh simple uh population model and this uh model is also known as logistic model. We'll discuss more about this model further time of duration. And the fourth is uh md²y/dt² = mg - αdy/dt.
[9:03]If you look at uh in this, if α is equal to zero, then it reduces to the uh differential equation given in two. And if α is non-zero, then it will turn out to be this. In fact, uh it is uh representing the same uh the uh motion of a particle falling from some uh height. The only thing is that now we are also considering that the there is some kind of resistance due to air, and that resistance is proportional to the velocity of the particle. So here, uh resistance means uh it is negating the motion of the particle. So in that case your motion of particle can be given as md²y/dt² = mg - αdy/dt. Here g represent the gravitational force. So in second and fourth, your g represent the gravitational force. So it means that fourth is the represent the differential equation uh which represent the uh motion of particle, when there is a some kind of uh resistance due to air. So these are some simple example of ordinary differential equation. Now uh we define certain basic things so that we can discuss more about differential equation. So first important part is the order of a differential equation. So order what is order of a differential equation? Order of a differential equation is the order of the highest order derivative present in the equation. So you consider equation and look at the highest order present in that particular equation. That is considered to be the order of the differential equation. For example, in the first, the highest order derivative present is only one that is dy/dt. So, this is first order differential equation. Similarly, this equation number third is also first order differential equation. While as uh your equation number four and five uh equation number four and two, the uh there are two derivatives present, one is uh dy/dt, another one is d2y/dt. And the highest order is d2y/dt. So it means that here the second order derivative is present. So, the order of the differential equation is coming out to be two. So it means that equation number two and equation number four is an example of second order ordinary differential equation. Whereas this equation number one and three is an example of first order differential equation. So first we try to know what is uh order of a differential equation. Then uh look at um uh the basic concept of uh of ordinary differential equation. So, as we have already discussed that an ordinary differential equation of nth order is defined as a relation, an equation between t, independent variable, y dependent variable and its high order derivatives. One or more high order derivative. So, any relation, any uh relation between t, y and y dash up to y and equal to zero, this is known as uh ordinary differential equation and the highest order present is given here, y and so this is an example of nth order uh differential ordinary differential equation, because number of independent variable is only one, that is t here. So here this y i where I is running from one to N represent the i-th derivative of the unknown function y. And here f is defined in some subset of Rn+2 and provides a relation between the N+2 variables that is t, y, t, y dash t, y double dash t up to y and t.
[12:42]So basically this one represent the nth order ordinary differential equation. But uh there is a small thing we we need to consider here. that here this function f may be a implicit function of t y and y dash. So here equation one may represent a collection of differential equation rather than a single differential equation. So here because of it may happen that your function f is defined in implicit manner, then uh this equation number one may represent more than one differential equation at a time. In fact, here in in this particular thing here y is representing a scalar valued function. So consider the following differential equation y'³ - 3t²y'² + 3yy' = 0.
[13:32]If you look at uh this as a equation, this is an a relation between t, y. So t here, y, y dash and uh yeah. So, basically it is an a relation between t, y and y dash. So, I can call this as a differential equation. But here, you can see that here uh y dash, this is an a implicit differential equation in terms of implicit relation between t, y and y dash. So what we try to do here, we uh try to see that this actually represent more than one ordinary differential equation. If you simplify this further, then I can write this as y'³ - 3t²y'² + yy' = 0. We can take out this y' common. So, y' = into this thing that is y'² - 3t²y' + 3y = 0. So, I can write this as y' = 0 or y' = (3t² ± √9t⁴ - 12y)/2. So, basically this simple one ordinary differential equation may give rise to your three different ordinary differential equation. So here, rather than considering this kind of a relation, we assume that to order uh to avoid the ambiguity, we assume that the given differential equation is solvable in terms of the highest order derivative. So, highest order derivative is here in this y dash. So it means that if it is solvable in terms of the highest order derivative, then we can rewrite the equation number one as in this manner that is y power n = g (t, y, y dash up to y power n - 1). So here, uh this in this way we we say that if y is scalar valued uh function, then this will represent a single ordinary differential equation, rather than considering the multiple multiple ordinary differential equation as a one uh group, we consider the normal form or canonical form. And from now onward, we assume that whenever we talk about ordinary differential equation, we are talking about the uh ordinary differential equation given in terms of normal form or canonical form.
[16:00]Now, uh consider a once we have we know the what is differential equation, then we try to know what is uh the solution of the differential equation. So, we define solution of the differential equation as follows: a function φ (t) is called a solution of two given in normal form on an interval I which is given as a comma b where b is bigger than a. And if it satisfy the following condition that φ (t) is defined on this interval. And it should be N times differentiable on this interval I. And second thing that φ (t) satisfy the equation two for each t equal to I. So, if we are able to find out such a function, we call such a function as a solution of the differential equation and given in normal form that is this kind of form. So, uh this is the definition of solution of the differential equation. And we in in all the problems we are trying to find out the differential solution of the differential equation in uh defined in terms of this definition. So the aim of the study of the ordinary differential equation is to find the unknown function represented in an explicit form. That is it should be that y is defined in terms of t using some elementary functions. What are elementary functions? Some example of elementary functions are these uh sine function, trigonometric functions, polynomials and logarithmic functions. So, it means that our primary uh aim is to represent yt in terms of uh t in terms of elementary using elementary functions. And in the absence of an explicit form, we need to study the behavior of solution by available analytical methods. So, in in case of when we we are not able to find out the solution given in terms of explicit form, we try to focus on the properties of the solution given in terms of implicit form. So, uh now um so we know what is the solution and what is the aim of the study. Now, we uh to start with the solution procedure, we classify our uh differential equation. So, the classification is basically based on mainly based on two categories. First is classification based on dependent variables, that is linear or nonlinear. So the classification based on dependent variable gives you a linear differential equation or a nonlinear differential equation. And the second one is the classification based on conditions, whether it is initial value problem or say boundary value problem. So, we'll discuss one by one what is uh these classification and how we can classify a given differential equation. So, first consider the differential equation given in normal form that is y power n = g (t, y, y dash, y up to y power n - 1). And if the relation G, this relation is linear in its argument. Argument here is t y y power n - 1, but the linearity or non-linearity will depend on the dependent variable and its derivative. So if G is linear in y, y dash up to y power n - 1, then the differential equation three is called a linear ordinary differential equation, otherwise it is called a nonlinear ordinary differential equation. So for example, if you look at this y dash + ky = 0, we can see that here your y y dash coming in a linear manner. So we call this a linear differential equation. And whereas the second problem that is dy/dt = y². Here if you look at, here y² is not linear. This term y² is not linear, so we say that it is a non-linear differential equation. If we simply say that linearity means that they are they present in the equation in a linear manner in the sense that they have only one power. And the coefficient of y or derivative of y may be only the function of t only, the independent variable. But if you follow this kind of definition, then it is very difficult to check whether this third equation that is y dash + modulus of y = 0 is a linear or a non-linear differential equation. Because here, it looks that y and y dash are coming in a linear manner. The power is only one. So, the procedure that checking that whether the y or the derivatives of y are present in the equation in a linear manner may not give you a proper definition of linear and non-linear differential equation. So, here uh to give a proper definition of a linear and non-linear differential equation, we use the uh concept of uh operator. And we say that if we define suitably what is the operator based on equation number three. And if the operator is linear, the corresponding differential equation is also linear. So, let us consider the following differential equation of order two, written in operator form. So, let us take an example and we have taken the example of differential equation of order two. And define L y as y double dash + p (t) y dash + q (t) y. And your differential equation is this that y double dash + p (t) y dash + q (t) y = r (t). So, basically I can write this that L y = r (t) is your differential equation. And where L y is defined as this that L y = y double dash + p (t) y dash + q (t) y. So here once operator is defined, now we check whether this this operator is a linear operator or not. If you look at here the notation this L y suggests that the operator L operates on a function y to give and gives this value. So output is y double dash + p y dash + q y.
[22:00]And input is your y. So, basically operator is basically what? An operator L defined on vector space V uh with the scalar field K to vector space V over K is said to be a linear operator on a vector space V defined on a scalar field K if it satisfy the following equality. That L of alpha x + beta y = alpha times L x + beta times L y where these x y is coming from vector space and these alpha beta is coming from this scalar field. Then if an operator satisfy this property is given in equation number four. Then we call this operator as a linear operator. And if it is a linear operator, and if it is associated with differential equation in this way, then this differential equation that is y double dash + p (t) y dash + q (t) y = r (t) is a linear differential equation. So here now let us look at here. So look at this y dash + mod of y. So you define L y as uh y dash + mod of y. And uh so it is an operator, it takes input value y and giving the output as y dash + mod of y. And then now look at the uh whether it is a linear operator or not. So look at alpha y1 + beta y2. And this is what? Alpha y1 + beta y2 dash + modulus alpha y1 + beta y2. And if you look at alpha of L y1 + beta of L y2, then it will give you alpha y1 dash + modulus of alpha mod of y1 + beta times there's a small problem here. Here it is alpha is outside. So here you can write it like this. And beta times y2 dash + modulus of y2 here. And alpha times L y1 + beta times L y2 is given by this. And you can check that uh though these terms are same that if you different if you simplify this is nothing but alpha y1 dash + beta y2 dash. But these term is not equal to modulus of alpha y1 + beta y2.
[24:45]So it means that here your operator L defined as y dash + mod of y is not an linear operator. So it means that the differential equation y dash + mod of y = 0 is not a linear differential equation. So we say that this is a an example of a non-linear differential equation. Similarly, we can consider the other example. For example, let us consider the second example. And here you define L y as uh y dash + a t y. Basically, when we check linearity, we are checking the linearity in terms of dependent variable and its derivative. So here now we write L y as y dash + a t y. And now we check whether this defined a linear operator or not. So you check that alpha y1 + beta y2 is coming out to be alpha times L y1 + beta times L y2, that is your right hand side.
[26:24]So, here your L y defined as y dash + a t y satisfy the property of linearity. So we say that this differential equation defined as L y = b t is a linear differential equation. So, this is the classification based on uh say dependent variable, and we say that since this differential equation is linear in terms of dependent variable and its derivatives, we call these kind of differential equation as um linear differential equation or non-linear differential if it satisfy the non-linearity. Now, uh let us look at the classification based on the conditions provided with the differential equation. So, consider the following differential equation y' (t) - α y (t) = 0, t ∈ R, which may represents the population growth model in a single species, as we have pointed out. Now, we may easily find out the solution of this, and the solution is given as y (t) = ce^(αt). Because it is a simplest differential equation we have already considered. And here c is an arbitrary constant.
[27:39]So, it means that if we vary our C, we'll uh have different uh different different solution of the equation number five. But our real model problem, basically this equation number five is coming through some kind of a real situation. So, it means that we are expecting a uh unique answer to that particular problem. So, it means that we have to find out rather than infinite many solution, we have to find out the unique solution. So, for that, we have to prescribe some kind of condition associated with the differential equation. So in this case, we may provide the conditions in terms of the initial conditions.
[28:30]And boundary conditions. Now we try to understand what is initial condition and what is boundary condition. So that we can understand with the help of some example. So first let us consider that we want to find out the population of the species at any given time t, provided that the population at time t naught was given as y naught. So, it means the initial condition, the condition associated with the differential equation is given in terms of that at t = t naught, your population is given as y naught. So, it means that y at t naught = y naught. And now we want to find out the y population at time t. And we say that if you look at different different value of this y naught and t naught, you may have a different solution of the same differential equation. For example, if I take y naught as zero, then you can get c = 0. So if I look at y (t) is basically what? y (t) = ce^(αt). So it means that if I assume that y naught is zero, that is 0 = ce^(αt) naught, then the only possibility that we'll get this equation valid, that c has to be zero. And in this case, when the initial population is zero, then y (t) will remain zero for all future time t. And if you replace y naught = 1. And in this case, 1 = ce^(αt naught). So, we can find out the value of c that is coming out to be c = e power minus alpha t naught. And your population is given as y (t) = e power alpha t minus t naught. Now in case of when t naught is replaced by zero and y naught is given by some value say y naught, then you can consider the we can calculate the value of c as y naught. And your population at time t is given by y (t) = y naught e power alpha t. So, here we can say that by defining different different condition at time uh at the initial point that is t naught, we may have different different possibility of solution.
[30:54]So, it means that the condition defined at one point or say initial point is known as initial conditions.
[31:07]And the differential equation associated with the initial conditions is known as initial value problem here. Now, look at the another example. Here we are again considering a very simple example y double dash + y = 0, which may be considered as a example originated from a motion of simple pendulum, y double dash + y = 0. And if we we we already know how to solve it and we can solve this and we can get y (t) = α sin t + β cos t. By the way, here I'm assuming that you know some uh how to solve some simple ordinary differential equation, for example, uh linear differential equation, exact differential equation, or uh say reducible to exact differential equation or higher order equation involving say uh constant coefficient, that I'm assuming that you know. So here uh once we know the solution, then if we define condition, since it is a second order, we need to fix uh two arbitrary constant that is alpha and beta. So we need two conditions. So let us define conditions as y of 0 = 0 and y of π/2 = 0. And if you do this, you can easily check that your alpha and beta both are coming out to be zero. For example, if y 0 = 0, then it is your alpha into 0 + beta. So this implies that beta = 0. Now y of pi by 2 is 0 = 0 implies that since beta is already zero, so alpha and sin of pi by 2, that is 1. So this implies that alpha is also zero. So it means that if we assign these conditions that is y0 = 0 and y pi by 2 = 0, then solution is now coming out to be non-zero and it is coming out to be y (t) = sin t.
[33:04]So, look at these two example and look at that in example one that is this example. Here your condition is given at point t = t naught here. Whereas in this example, your conditions are given at two different point, that is zero and pi by two. So it means that condition given at the same value of t are known as initial conditions.
[33:33]So condition given at one point is then known as initial condition. Whereas the condition defined at two, generally it should be the end point of the uh interval or more uh different points are called boundary conditions. Right? So this is the classification based on uh say dependent variable, and we say that since this differential equation is linear in terms of dependent variable and its derivative, we call these kind of differential equation as um linear differential equation or non-linear differential if it satisfy the non-linearity.
[34:11]Now, so uh so far we have discussed uh the um classification and definition of uh uh differential equation, solution and um um the required information related to this differential equation. And in next lecture, we'll continue from this. So here we'll stop and we'll conclude this lecture. Thank you very much for listening us. Thank you.
