 Alright, so what is a differential equation? A differential equation is basically an equation which involves the independent variable x, the dependent variable y and the differential of the dependent variable with respect to the independent variable that is dy by dx and dy by dx can be in various orders. To give you an example of a typical differential equation, something like d2y by dx square equal to minus 3x. So basically this is a differential equation. We can have a y as well here. So basically this is a differential equation which involves an independent variable, dependent variable and a particular order. There may be various orders of the differential coefficients involved. This is called a differential coefficient of second order. So any kind of equation which involves a relation or an equation involving x, y and various orders of the differential coefficient of y with respect to x and of course constants can be there is called a differential equation. The use of differential equation is tremendous. If you are talking about physics, you use this in the simple harmonic motion. You can use this in the RC circuits. You can use this in chemical kinetics. In the last part of the chapter, when we are doing the application of differential equations, we will take a lot of examples where we will show you the physical relevance of differential equations for you. All right students, now let's talk about the order and degree of a differential equation. Now order of a differential equation is nothing but the highest differential coefficient of y with respect to x which occurs in that differential equation. Are you sure about that? To give you some examples of what I mean when I say highest differential coefficient of y with respect to x, let us take example number 1. In this you can see we have d3y by dx cube raised to the power of 2 by 3 equal to dy by dx plus 2. In this particular differential equation, you would realize that the highest differential coefficient of y with respect to x is 3. This means it has y has been differentiated a maximum number of 3 times with respect to x. Please note this power has no relevance while deciding the order of the differential equation. So please do not take into account that since there is a power of 2 by 3 it should become a d2y by dx square. No, that doesn't happen. In this case, we say the order is nothing but the maximum number of times of the derivative of y that has happened with respect to x in that equation. So this is order 3 differential equation. So we say this is a differential equation of order 3. Let us take example number 2. You can guess, you can take a call. You can see that dy by dx, d2y by dx square is equal to x l and dy by dx. So out of these two the highest order is given by this term because here y has been differentiated twice with respect to x whereas in this expression y has already been differentiated once. So out of the two since this is the higher, we say that the order of this differential equation will be nothing but 2 because 2 times derivative has occurred here. So this is a differential equation of order 2. Again repeating a very, very useful information. When we are deciding on the order, we just look into the entire expression and pick out that term where the derivative of y with respect to x has happened maximum number of times. That decides the order of the differential equation. All right, so now let us understand degree of the differential equation. Now this aspect of degree is slightly critical. So you have to be very, very attentive while listening to whatever I say now. The degree of a differential equation as I have already written on the board, we can read this along. It is the exponent of the highest differential coefficient. What is exponent? Exponent means the power. It is the power of the highest differential coefficient. Now in the previous video we had talked about the order of the differential equation where I had taught you how to identify the highest differential coefficient. That means what is the maximum number of times the derivative of y has occurred with respect to x. So once you have identified the highest differential coefficient which is nothing but, which decides the order of the differential equation, the power of that term, that is the exponent of that term, when that differential equation has been expressed in the form of a polynomial in all the differential coefficients. Now this is slightly critical. What do I mean when I say when the differential equation has been expressed as a polynomial, the word polynomial in all the differential coefficient. Now let us go back to our class tenth. What do we understand from a polynomial? Now if you recall a polynomial function is something where you have terms like this. So we have terms of this nature. It can go all the way till constant. Where one thing has to be kept into care is that the power of these variables, these powers should be whole numbers. That means none of the powers should be fractions, none of the powers should be negative integers. There must be whole numbers. 0 is acceptable, national numbers acceptable, but none of the powers on the variables which is x in this case must be fractions, must not be, there must not be negative integers. In a similar way when your differential equation has been expressed in such a way that all the differential coefficients no matter whatever order they are, if their power has been made whole numbers, then only we can decide on the degree of the differential equation. Are you sure about that? Let me show you with the help of an example. Let's say I have a differential equation like this d3y by dx cube whole to the power of 2 by 3 equal to dy by dx plus 2. I have taken a very similar example in the previous video also. The order of this differential equation is what? It is 3. This decides the order. So it is an order 3 differential equation. So order 3. So there is no doubt about the order. But wait a minute. When I am deciding about the degree, have I expressed this in a polynomial of the differential coefficients? No. We have not expressed in the polynomial of differential coefficients because the power on this differential coefficient is 2 by 3, which is not allowed in a polynomial. So in a polynomial x should not have a fractional power. In a similar way, this differential coefficients involved in this expression should not have any kind of a fractional power. So this is having a fractional power which is not acceptable. So does it mean that I cannot find the degree of this differential equation? I still can but only after getting rid of this fractional power. How do I get rid of this fractional power? It's simple. I will take the cube of both the sides. So I will cube both the sides of the equation. So when I cube both the side of the equation, I will get d3y by dx cube whole square equal to dy by dx plus 2 the whole cube. Now having done this, now what is the highest or exponent of the highest differential coefficient? The highest differential coefficient is d3y by dx cube. No doubt about that. The exponent of that is 2. So having converted it to a polynomial, the highest power of or the power of the highest differential coefficient is 2. So hence this differential equation will have a degree of 2. So this will be a differential equation whose degree will be equal to 2. Hope it makes sense to you. So while finding the degree, two things you must keep into account. What is the coefficient or what is the exponent of the highest order? And you must convert that differential equation into a polynomial, a polynomial form in all the differential coefficients. If it is not, degree cannot be found out. We say degree cannot exist. A degree does not exist for that particular differential equation. All right friends, now let us solve the questions which we just now displayed on the screen. So we will start with the first question. We need to find the order and degree of this differential equation. Now finding the order and degree of this differential equation is very simple. In order to find the order, we just see which is the highest differential coefficient of y with respect to x occurring in that differential equation which clearly is this d2y by dx square term. So d2y by dx square term is the highest differential coefficient of y with respect to x in this video. Hence order becomes 2. So finding order does not take much time. You just have to identify the highest differential coefficient and your job is done. Now while finding the degree of this differential equation, we first need to convert this into a polynomial form in all the differential coefficients. In the present stage, it is not because if you see the power of d2y by dx square is half because it is under root. And in this case, we have dy by dx plus 3 whole subjected to the cube root power. So if I have to write in terms of power, I can write this as d2y by dx square to the power of half equal to dy by dx plus 3 whole to the power of one-third. And in a polynomial, we cannot have these powers. So these powers are strict no-no in polynomials. So in order to convert it to a polynomial form, I need to apply some tricks. I need to first ensure that I make these two numbers as whole number powers. And for that, we need to multiply these powers by the LCM of 2 and 3. So LCM of 2 and 3, we all know LCM of 2 and 3 is 6. So what I will do? I will subject both the side of the equation to the power of 6. So when I do to the power of 6 on both the sides of the equation, I would realize the left side becomes d2y by dx square whole to the power of 3 equal to dy by dx plus 3 whole square. Having done this, I realize now each of the differential coefficients involved in this equation will be subjected to a whole number power. That is what is meant by converting the differential equation in polynomial form. Having done this, now I identify what is the exponent of the highest differential coefficient. Undoubtedly, the highest differential coefficient is d2y by dx square. The power or the exponent of this is 3. So we say degree of this differential equation will be 3. So this is in order to degree 3 differential equation. So this is how we solve question number 1. All right friends, coming to the second question. Now finding the order again is a very simple task in this. We just have to identify the highest differential coefficient of i with respect to x. Undoubtedly, it is this term d2y by dx square. So d2y by dx square corresponds to an order of 2. So the order for this becomes 2. Finding order is the most easiest of both the activities. Now how to find the degree? Now for degree, as I already discussed in the theory, we need to convert this into a polynomial form. Now this sine term here creates a problem. Have you seen any polynomial having a sine term in it? No, of course not. A polynomial only has different powers of x and those powers must be whole numbers. So there is no sine term, there is no log term, there is no exponential term. So as to say there is no transcendental functions involved in a polynomial. Now here being one, I cannot say that this is in the polynomial form. But hold on, can I get rid of it? If yes, then try to. If not, then we cannot find the degree for this particular differential equation. Or we say the degree for this differential equation is not defined. So in this case, no matter how much ever we try, we will not be able to eliminate the sine form from this particular differential equation. Or in other words, we will not be able to convert this to a polynomial form in the all differential coefficients. So hence for this particular differential equation, degree will not be defined. So we say degree is not defined. All right students, coming to the last question, the third question. In this case again, we have to find the order and the degree of this differential equation. Order again is a very simple activity. We see only dy by dx, that is the only differential coefficient existing in this problem. So it's an order one differential equation. Now what about degree? So order is one. What about degree? Now for degree as I told you, we need to convert it into a polynomial of differential coefficients. Now many students get confused over here. They say that it is not a polynomial, it is not in the polynomial form. So how are you finding the degree for it? Now please note the under root sign on the independent and dependent variables are allowed. So in this case, we have an under root on 3x plus 5. It does not involve any differential coefficient. Had the under root been on some differential coefficient, let's say the root have been on dy by dx or some negative powers would have been on dy by dx or a non-whole number power should be on dy by dx, then in that case, yes, I understand it is not in the polynomial form. But in the present scenario, this is perfectly allowed. We can have an under root or a radical power or a negative power on the dependent and the independent variables. But it should not be on any differential coefficient. So in this case, degree is simple. It's just the power on this highest order differential coefficient, which is one in this case. So degree is one in this case. Not only that, if I just do an extension of this question, for example, if I do dy by dx equal to under root of 3x plus 5 plus, let's say, sign x plus, let's say, log xy, right? What is the degree for this? Degree of this will still be one. Please note, this sign, this root and this log term has no bearing on deciding the degree. So it is still a polynomial form and degree can be found out. So the whole idea is that the differential coefficients involved should be in the polynomial form. That is, none of the power on the differential coefficient should be a negative or a fractional part. On the dependent and independent variable, we can have sign, log, negative powers, whatever you feel like, can occur on the independent and dependent variables. So for this, the degree still is equal to one.