 alright so in this video we are going to talk about formation of differential equation and before I start talking about it let's let's pause and ponder why at all we need to form a differential equation right what was the necessity that a differential equation should be formed right now to explain this concept let's go back to the origin of the formation of equations let's forget about differential equations let's just talk about why at all there is a need to form equations right so when we say the equation of a straight line is y equal to let's say x plus 2 right what do I mean by it now let us understand this because if you are able to answer this we'll be able to answer the fact why at all a differential equation is formed what was the need for a differential equation now when the in the beginning of the geometry when we wanted to represent some set of points right or we wanted to show a path for example if I say that I am treading on a path whose coordinates are 1, 3, 2, 4, 3, 5, 4, 6 and so on and so forth right it's very difficult to actually write those infinitely number of points which actually I am treading okay so instead of writing these points as one by one we can actually say that I am treading on such a path where the x coordinate and the y coordinates of the points on which I am walking are related to each other by this relation so if you choose x as 1 y automatically becomes 3 which is this point correct if I take x as 2 y automatically becomes 4 which is this point so instead of seeing all these points which are infinitely so many number I can simply say that I am walking on a line whose equation is y equal to express 2 isn't that a much easier way to express the path which I am walking so that straight line instead of saying I am walking on a straight line which is made up of point 1, 3, 2, 4, 3, 5, 4, 6 it's simple to say I am walking on a line whose equation is y equal to express 2 isn't it so this is nothing but a relation which connects the x coordinate this is nothing but your abscissa this part and the y coordinate which is nothing but the ordinate of this point so 1 and 3 are related to each other in this fashion just like 2 and 4 are related to each other in this fashion in a similar way even for differential equations right if I asked you this question dy by dx is equal to let's say 2x right so this is actually a differential equation correct now if I say this is a differential equation which is valid for the curve y equal to x square correct everybody knows when we find the derivative of x square with respect to x man so will be 2x somebody else can say it is also valid for a curve x square plus 1 right somebody else can say it is also valid for the curve x square minus 5 so in general I can say the same differential equation is valid for all those curves which are of the type y equal to x square plus c where c is an arbitrary constant you can choose your c anything you want and this differential equation will still be valid for that so in order to explain differential equation in one line I will just say differential equation is nothing but a representation or I would say an aggregate representation of a family of curves that means a set of curves which belong to the same species so here if you see it's a parabola so all the parabola which belongs to this species is collectively categorized as this differential equation right just like all the set of points which satisfy equation that your y coordinate is 2 more than the x coordinate is collectively specified as this right so this brings me to a very very important concept in differential equation called the solution of a differential equation right now solution of a differential equation I will I will talk about this in the later part of this video but as of now just to give you a heads up on this this particular curve is a solution to this differential equation correct because this satisfies this differential equation or in other words this differential equation has been this differential equation has been formed for this curve or has been formed from this curve okay so we are going to talk today about how to form a differential equation once you know the solution so once this has been mentioned to you how do you form a differential equation now it may sound very easy to you you just have to differentiate it and get the answer but however there are some set of rules which you need to keep in mind while you are forming differential equations which I'll be talking in some time alright students now we'll talk about how to form a differential equation in the previous section we had talked about why at all we need a differential equation and what does it represent now we'll talk about how a differential equation is formed now you can see on your screen that I had have listed out the three steps that we need to follow in order to form a differential equation right now a differential equation is formed for a family of curves right so basically a family of curves will be either given to you or a situation will be listed out from where a family of curves can be generated right so to keep the case simple I will give you a family of curves and then we'll learn how to form a differential equation from it okay so let us talk about an example where I have given you a family of curves as y square is equal to 4a x plus b okay anybody can guess here that I am talking about a family of curves which represent parabola right here your a and b are arbitrary constants a and b are arbitrary constants okay so your a and b are like parameters which keep on changing and as it changes they keep on represent representing different parabolas however all those parabolas belong to the same family okay now let us follow the steps which I have listed out the first step is I need to differentiate this equation or the equation of the family of curves as many times as the number of arbitrary constants involved okay now this has a very very important relevance this says that your order of the differential equation that is going to be formed should be equal to the number of arbitrary constants involved in that curves equation I'll repeat this again the order of the differential equation that is getting formed should be equal to the number of arbitrary constants involved in that equation so in this equation we can see there are two arbitrary constants involved one is a another is b so this will result into a differential equation whose order will be 2 why because they are two arbitrary constants so I can write here order should be equal to the number of arbitrary constants involved this is very very important so the order of a differential equation that you are going to get form should be equal to the number of arbitrary constants alright now how do we form the arbitrary constants how do we form the differential equation in the differential equation we differentiate and step number 2 we tried to eliminate the arbitrary constants in that given equation so while you are differentiating you will keep on getting some equations which along with the original equation you can use to eliminate the arbitrary constants so a and b have to be finally been removed okay after that the element that is whatever is left after having got rid of the arbitrary constant will be the required differential equation which we need so let us apply these three steps on this problem so that things are clear let us differentiate once so what is the derivative of y square with respect to x we all know chain rule so derivative of y square with respect to x will be 2 y dy by dx right derivative of 4a x plus b will be nothing but 4a right we can get rid of the factor of 2 from both the sides so I can drop this two factor here and write it as equal to 2a right now I have already got rid of b now the only thing which I need to do now is get rid of a simple to get rid of a I will differentiate it one more time because I have a liberty to differentiate as many times as the number of arbitrary constants so since they were two arbitrary constants I have a liberty to differentiate it one more time because one time I have already done so let us differentiate both sides again with respect to x now care should be taken while differentiating this with respect to x when we are differentiating y dy by dx I'll be using which rule the product tool exactly so I'll be using this and this as two functions u and v and I'll be applying the product rule of differentiation so let us take y times derivative of dy by dx now guys please note the derivative of dy by dx with respect to x is nothing but d 2 y by dx square so I'll have y d 2 y by dx square okay plus keep dy by dx as such into derivative of y derivative of y we all know is dy by dx so I'll have again dy by dx coming over here isn't it and to a derivative we all know is going to be 0 a being a constant towards an arbitrary constant but it's a constant so its derivative will be equal to 0 right so simplifying this further I can say y d 2 y by dx square plus dy by dx whole square is going to be 0 this is the equation which is actually the differential equation which corresponds to this family of parabola right so this is your answer which we need to obtain so again repeating all the steps in in the order in which you have to follow you can you have to first start with differentiating the curve you can differentiate it one by one but you have to differentiate it only as many number of times as the number of arbitrary constants involved so if the number of arbitrary constants are one you can differentiate it only once if there are two you can differentiate it twice right it should not happen that there's a one arbitrary constant and you're differentiating it two times that should not occur right and while you are differentiating you'll keep on getting equations right using the equations along with the original one right many people many students make a mistake they forget to use the original equation given to you no you can use the original equation along with the equations that you are getting while you are carrying out the process of differentiation and use them all to eliminate the arbitrary constants by hook or by crook I have to get rid of A and B right in this problem I have to get rid of A and B so all the arbitrary constants have to be got rid of after having got rid of all the arbitrary constants the element that means the remaining remaining thing for example here this is the element this element represents the differential equation for this family of curves in other words this differential equation will be satisfied by this curve or this differential equation has been formed for this family of curves just two way of looking at the same thing right hope this process of forming the differential equation is clear to you so hope you have tried this question on your own so now we'll be solving this question for you as you can see this is a family of curve y equal to a e to the power 3x plus b e to the power x right why it represents a family why not a single curve because we have A and B as arbitrary constants so these constants will keep on changing and as they change they represent different different curves of the same family okay now we have to form a differential equation which satisfies this family of curves or in other words the other interpretation is find the differential equation which is formed from this family of curves okay now since there are two arbitrary constants I have a liberty to differentiate it two times correct so let us start by differentiating it once first so dy by dx will be nothing but a times the derivative of e to the power 3x now derivative of e to the power 3x will follow the chain rule so it will be e to the power 3x times 3 so let me write 3 over here okay plus derivative of b e to the power x will be b e to the power x isn't it okay now here if you see this this equation we have not been able to eliminate any of the arbitrary constants involved so A and B are still involved all right now what you'll do in this case now I will try to reduce the number of constants here for example this b e to the power x I can replace by y minus a e to the power 3x right so what I mean to say is that I can replace b e to the power x as dy by y minus a e to the power 3x all right so I am going to replace it over here so I can say dy by dx is equal to 3a e to the power 3x plus y minus a e to the power 3x right on slight simplification I can club these two terms and write it as dy by dx equal to two times a da e to the power 3x right now let us differentiate it once more time so once more if I differentiate I'll get d2y by dx square equal to now this will be 2a e to the power 3x into 3 so I can say this into 3 okay plus dy by dx correct so I've used the chain rule over here and y derivative is dy by dx and of course dy by dx derivative is d2y by dx square okay now again this term is appearing over here right so what I will do now I will replace I will replace 2a e to the power 3x 2a e to the power 3x with dy by dx minus y dy by dx minus y okay where here at this location all right so what I'll get now d2y by dx square will be equal to 3 times this term which is nothing but dy by dx minus y again I have a plus dy by dx waiting outside okay now as a result you see that we have got rid of all the arbitrary constants here do you see any a b in this equation no we don't so we have got rid of arbitrary constants and at the same time I have taken care that I'm differentiating it only twice why because there are two arbitrary constants so summing this up we can conclude that the differential equation formed for this family of curves will be d2y by dx square equal to 4 dy by dx y4 because there's a 3 dy by dx here and 1 dy by dx is over here minus 3y okay so this is the differential equation for the family of curves given by this okay so by this time you would have realized that formation of differential equation is not a straight forward activity right you have to apply your you know all all sorts of strategy to get rid of a and b right and you're working under the constraint that you cannot differentiate it as as many number of times as you want right you have a limitation that your derivatives or the number of times you are differentiating should not exceed the number of arbitrary constants alright friends in this question this is a slightly differently framed question where you have not been mentioned the family of curves site so you are instead asked to find the equation of a family of curves and then obtain a differential equation out of it right so the question says you have to write the differential equation for the family of curves which are family of circles which touches the x axis at origin right now let us understand this so let's say this is our x and the y axis is okay so there's a circle which touches the x axis at origin so I can have a circle like this right I can also have a circle like this right I can also have a circle like this so they can be so many circles which I can draw which will touch the x axis at origin right and we have to write a differential equation which signifies all these family of circles correct so the first thing is I need to write down a family of such circles equation right how do I write that very simple I can assume the center of us this circle to be let's say 0 comma a y 0 comma a obviously when it is touching the x axis at origin that means a line perpendicular from the origin should pass through the center of the circle and any line perpendicular to the x axis has to be the y axis right and it is passing through the origin so any line which is passing through the origin and perpendicular to the x axis has to be the y axis so in other words the center of the circle lies on the y axis so I can assume the center to be some 0 comma a correct and not only that you will also observe that the radius of this circle will also be a now what is a a some arbitrary constant why because I'm not sure which circle I'm talking about whether I'm talking about this one or this one or this one or others to come so this a is something which is varying and because it is varying I'm calling it as an arbitrary constant right because it can be anything it can be arbitrary right so the equation for such a circle whose center is that 0 comma a and set and radius is a can be written as x minus 0 whole square plus y minus a whole square equal to a square right so you all have done circles in class 11th so everybody is familiar with the equation of a circle whose center and radius is known correct now so I have one arbitrary constant here please note there are no no two arbitrary constants only one arbitrary constant is there which is a so a here is an arbitrary constant and therefore I have a liberty to differentiate it how many number of times once because there's only one arbitrary constant many students make a mistake they think there's an arbitrary constant here there's an arbitrary constant here no and you have only one arbitrary constant in this particular problem so I can differentiate it once so let us do that what the derivative of x square with respect to x we all know it's going to be 2x right derivative of y minus a whole square is 2 y minus a times dy by dx correct the derivative of a square with respect to x is going to be 0 okay now we can drop the factor of 2 from everywhere so let us drop the factor of 2 from everywhere so 2 2 can be dropped off right now I need to somehow use this equation and the given equation so basically let me name it I can use 1 and 2 right in tandem to eliminate what eliminate the arbitrary constant which is a right so how do I do that very simple I can write this as a minus y dy by dx equal to x okay so all I have done I have brought the x on this side which is minus x I change the sign both the places in other words a minus y is nothing but x by dy by dx okay in other words in other words a can be written as y plus x by dy by dx okay right so we can simplify it further to x plus y dy by dx divided by dy by dx alright now having obtained a in terms of x y and dy by dx I will do a simple activity I will replace this back in equation number one okay so I will substitute I will substitute a let me call this as the third equation I will substitute a from one okay so what I am going to do now I am going to substitute this value of a in one right but before that let us simplify it if I simplify it I will get x square plus y square minus 2 a y equal to 0 okay having done this let's substitute the value of x sorry the value of a so the value of a from there I can write it as 2 y times x plus y dy by dx whole divided by dy by dx equal to 0 correct now it's a good practice that you should not leave a dy by dx in the denominator so let us multiply throughout with dy by dx so which gives you dy by dx times x square plus y square minus 2 x y minus 2 y square dy by dx equal to 0 right I can take dy by dx common over here so which finally gives me x square minus y square dy by dx minus 2 x y equal to 0 so this gives you that final element that is the differential equation that satisfies all these family of circles so this satisfy all the curve which is in this family of circles right so this is the differential equation where you have seen that I have not differentiated it more than once so you can see the order of this differential equation is one remember order it's the highest differential coefficient that occurs so it's one here differential coefficient means dy by dx and all the arbitrary constants here we had only one arbitrary constant a it has got eliminated right so this is our answer for this problem it fits so when this question was asked in the exam many students wrote the answer for this as 4 right the order of this differential equation is 4 right and if you have done the same please let me tell you you are wrong right why it happens why this differential equation or why the differential equation that is to be formed for this family of curves does not have an order of 4 right it's not a simple rule that we count the number of arbitrary constants 1 2 3 4 and write the answer as 4 right in order to find the order of the differential equation whose solution is this we need to see what is the effective number of arbitrary constants involved which cannot be obtained by just counting the number of arbitrary constants written in that written in that curves equation we need to see effectively how many arbitrary constants are involved now in this question if you see it very closely these two arbitrary constant can be actually be treated as one arbitrary constant so even though it is C1 plus C2 effectively it is just one arbitrary constant which is involved right so we can write this first of all as some arbitrary constant a so I have clubbed C1 and C2 and written it as a e to the power x okay here also if you see I can write this as C3 e to the power x into e to the power C4 okay now C3 into C4 they will also multiply it to generate only one arbitrary constant right we cannot write this as two arbitrary constants so this is a constant agreed this is a constant agreed their multiplication will effectively the word is effectively will generate only one arbitrary constant so I can club them and write it as e to the power x not only that e to the power x if I take common I can write it like this right now a and b both arbitrary constants so when these two arbitrary constants are joined I'll get another constant right so I can call this as another arbitrary constant let's say capital C right so in short if you realize the effective number of arbitrary constants involved are only one right so this particular family of curve or this is a solution to a differential equation whose order should be equal to one right so the answer is one in this