hello students my name is Dr gendra purohit and you are watching our YouTube channel where I upload videos for engineering mathematics BSC if you are preparing for any competitive exams where higher mathematics is asked our YouTube channel is very helpful for you so today I'm going to explain how to find the particular integral in a difference equation normally our answer is CF plus pi right so I explained how to find CF in the previous video If you haven't seen it go to it Tab and watch today I will tell you how to solve a particular integral indifference equation in numerical analysis so let's get [Music] started so let's discuss this whenever we have any kind of difference equation it's linear difference equation and here we have function FN so first we will convert this into its auxiliary equation and f e is its auxiliary equation we will take this to the denominator normally we have differential equation we know that we can write it like this for example let's say we have function PX so what we do is PX upon we have FD and then here we will find its particular integral the concept is same here whatever auxiliary equation we have we will take it down and then we will solve according to the function now I will try to explain function one by one first of all we'll talk about the a power n type of function if the function we have is a power n type so what will we do in this we have an auxiliary equation for e and here we have YN and suppose this is a power n so students YN is equal to a power n 5 e students wherever there is an e we will put a what will we put here we'll put a but a that we have should not be equal to zero now you might think what if a becomes Zer in that case what will we do so I will explain a trick though the formula that we have is if denominator becomes zero then we have e minus a power m f e and if upon putting a denominator is becoming zero and this is power M then when we differentiate it m times we get M factorial and when we do its derivative n * n comes in front then n minus one we do its derivative till M minus one of this a right you can easily remember this formula by this concept this trick what will we do here if we get 0 ones then this n will come forward and this will be a power n minus one so what will we have here 5- e okay so we will put 5- e here now what we will do here again we will use YN so n a^ n minus 1 it will be 5- a it's the same thing again value of 5- a should not be zero we will keep doing this process until we get something which is not zero here I am going to take an example in front of you like see here we have this example whereby n + 2 - 4 y n + 1 - 4 YN is equal 2^ n okay what are we going to do with this we we will write e s YN - 4 e YN - 4 YN is equal to 2^ n right so here it will come e 2 - 4 e - 4 into YN is equal to 2^ n is it clear so we will take this to the denominator so YN is equal to 2^ n upon e 2 - 4 e - 4 I am Telling You its particular integral you might wonder what will be its CF so I will tell you about it so to find its CF it would be e s - 4 e - 4 is equal to 0 I explained this in the previous video If you haven't seen it you can go and watch it it will be whole square of eus 2 is equal to 0 so here the value of e will be 2.2 right and here CF will be C1 + c2n 2^ n is it correct okay we have its CF end then we will find its particular integral so wherever there is e we will put two there and as we put two then it will become zero since it is zero then what will we do here and will come forward 2 ^ of n - 1 if we do its derivative it will be 2 e - 4 again put 2 here so 4 - 4 is again 0 we will take its derivative again right so here we will get n into N - 1 2 ^ of nus 2 so here in denominator we get 2 so this what we have here this will be its particular integral then the final answer will be RPI so the final answer we have is YN is equals to CF + PI right so what will be our CF it's C1 + C2 n 2^ n + n into N - 1 upon 2 and 2 ^ of N - 2 so it will be our answer right students so this here what is this it is particular integral we do it like this so let's move to the next concept if we have cos or sign here just like we have a differential equation it's same concept so here we have cos k n and sin KNN and we have to find its particular integral if you remember so what we do in this differential equation that fdy is equal to cos a X okay we will take this FD down this is about differential equations so that you can learn from it very easily this will be a x and in denominator minus a square whatever the angle is it is squared and subtracted same concept applies here nothing new so don't worry about that right so we have this differential equation right look here we have YN + 2 - 7 y n + 1 + 12 YN is equal to cos n okay so here we will get this e s YN N - 7 e y n + 12 YN is equal to cos n is it clear to you okay first we will do e s - 7 e + 12 and you can take out this YN and this will be c n first we will calculate CF and to find CF what will we do E2 - 7 e + 12 is equal to 0 so look here it will be e - 4 into e -3 these are its factors okay if we solve them then the value of e will be 4 and 3 so the CF will be C1 3 ^ n plus C2 4 ^ n here this will be the value of CF now we talk about the particular integral so this will be YN is equal to this will be cos n upon e s - 7 e + 12 the coefficient of n is 1 it's cuse n so coefficient is 1 wherever there is e Square it will be -1 sare so here its value will be cos n so this is -1 s - 7 e + 12 if you write this it will be this will because n upon here we will have - 7 + 11 I will do one thing take this minus out as it is common this will be 7 eus 11 right now what happens is when we talk about differential equations we rationalize it that means we multiply and divide 7 e + 11 both up and down students we multiply up and down by this 7 e + 11 so that we get e Square here so here we will get YN is equal to 7 e + 11 and here it will be cos n and we will divide it by here at the denominator we will have please look carefully - 49 e s - 121 okay here at place of e square if we put -1 Square it will be - 49 so - 49 - 121 is - 170 what will come here 170 right so this will be 170 in denominator so this e we need to convert this here and listen students we know that D is equal to e minus I right so the value of e here it will be D plus I so wherever there is e we will place D plus I there so that things are easily solved so it will be D plus I right and + 11 so here we will have cuse and right students now here we will get YN is equal to 7D + 7 and 11 so this will be 18 right it will be multiplied by cos n upon here we already have 170 here derivative of cos n can you see D here d means d and d means derivative right the derivative of cos n here it will be sin n which will be negative and then here we will have + 18 cos n divided 170 right students so we got its particular integral now then the final answer we will get is CF plus pi so we can very easily do this so next we'll see if the function we have is of n power P type then in that case what will we do whenever we have n to the power P type of function then we need to First convert it into factorial notation now question will arise in your mind what is factorial notation if you have seen my videos I had uploaded a video before this means a few days before this video where I explained about operations I explained the different operators that we have I explained all of them I also explained factorial notation but I will tell you a little bit about it if we have any factorial notation let's say we have n Cub if you want to write this in factorial notation we write it as a n to the power of 3 plus here be n the^ of two plus we have this CN the power of 1 and then here here we will have D okay convert this into factorial notation see how will we do it so here we have n Cub it means n into n -1 into N - 2 + B okay so this comes as n and 2 N - 1 + c n + D so we do it like this here right now what we do here is compare these values so you are seeing the coefficient of n Cub here is 1 and here the coefficient of n Cub is a clear so the value of a here is one okay if you look here the coefficient of n sare so here if we talk about the coefficient of n Square what it will be if you see here I will multiply this so that you can understand it better here we get n² and - 3 n and + 2 okay so the N square that we will get it will be Min - 3 n Square so this will come as Min - 3 a and here we will have B right here its value will be zero why is it zero because there is no n Square here right now we will talk about the coefficient of n it will be two here a is getting multiplied so this will be 2 A in that case what will we get here it will be 2 a - b + C = 0 right now discussing about constants so constant is not here the constant that we are getting here I think only D is there right d so this D is going to be zero anyway so I think this is anywhere zero clear now what we do A's value is one and if you put this A's value in it then B's value will be three now we got the value of a and b now put them here this will be two this is - 3 plus C so from here the value of C will be 1 okay put it here so here the factorial notation of n Cube will be value of a is 1 so here its value will be n ^ 3 plus B's value is 3 and here we will get 3 n ^ 2 and C's value is n here okay when you simplify this it will ultimately be equal to n Cub so this is how we convert it to factorial notation this is needed later I have told you this earlier in previous video right now we are talking about that ultimately whenever we have n power P type you must have seen whenever we solve a differential equation if we have anything of type x^ n then what we do here this x^ n upon FD then we take this up and solve with the binomial expansion right similarly we will be using binomial expansion here but first you have to converted shift operator into D then you can use this concept here let me show you how I will try to explain this to you please look here and try to understand okay we have have this difference equation given here the difference equation that we have is YN + 2 + YN + 1 right plus YN is equal to here we are getting this as n Square so first of all what we will do is convert this n square into factorial notation so how do we do this tell me this n square is equal to a n 2 + BN 1 + C what will we write this as we will write it as n² is equal to a n into n -1 + b n + C first coefficient of n Square value of a is 1 then we will look at please Focus here then we look at the coefficient of n here coefficient of n is minus a and here it's B is equal to zero we put value of a here then value of B will be 1 and C is zero so value of a and b is 1 factorial notation of n Square will be n ^ 2 + n ^ 1 now you will ask how does it come so when you open this and write this n into N - 1 + n then you will cancel this it will be n sare is it clear in this way we convert this into notation so that we can easily do its derivative let's get to the point here we will write this as e ² YN + e YN + YN now if we convert this into notation this will become N2 + N1 is it clear this is the value that we will have e sare + e + 1 and this will be YN to the^ of 2 + 1 right now what should we do first of all we'll find the CF once anyway we need it because whenever you are asked to solve it you need to find the CF and set it to zero right so the value of e will be b - 1 + - b s which is 1 - 4 upon 2 is it clear from here we get the value as -1 +- < tk3 Iota / 2 so here we are calculating the value of our am I right here Alpha that is given I had told you in the previous video about CF write the imaginary and real Parts separately so the value of R here will be Alpha square + beta Square so this will be 1X 4 plus 3x 4 so its value will be 1 then we will calculate Theta it's tan inverse y by x value of y is < TK 3x2 - 1X 2 so we will get tan inverse minus < tk3 right minus < tk3 so the angle we get here will be < - < by 6 right what will we get < - < by 6 okay we know that value of tan < tk3 is < by3 so the value we will get is < - < by 3 because it's negative if we solve it we get 3 and 2 from here we get its value as 2 piun by 3 so the answer that we get is CF this is CF so here we get this as our power n and C1 cos n theta plus C2 sin n Theta so so the value of R is 1 so it's 1^ n this is CF here right and this will be C1 cos N2 < by3 and C2 sin 2 N < by3 right so this will be CF okay let's talk about particular integral in particular integral what do we do first we convert this e to D so bring it down this will come as YN and here we will get n the^ 2 right we will get this now divided by take this down so we know that wherever the value we have that is D is equal to e minus 1 so the value of e is D + I right so we will apply that here okay so wherever there is e put whole square of D+ I plus here this will be D+ I and plus this is 1 we can also put I here right in factorial notation we have n ^ 1 right we call it factorial 2 or factorial 1 1 so it will be factorial 2 and factorial 1 divided by okay if we simplify this here we'll get d square and then it's two D and D so we have three D here and then this is 11 one so we have three here so like we do in differential equations we take three as common take it up and then we will inverse it YN is equal to here we will get 1 by 3 and then this will come as 1 + D sare + 3 D divided by 3 going up it will be inversed and this will be factorial notation of this right so here we will get its value now we will expand this and I am writing its expansion there so you can understand it very easily right students I think you have understood this point right so I will do this here we have YN is equal to open this we'll get 1x3 apply binomial expansion so here it will be 1 - d square + 3 Del by here we have 3 are you getting it plus whole square of d square + 3 D by 3 right students therefore as you can see what will we get here it will be n² + N1 this is its value so here we will have the value of y n as 1X 3 1 minus and here it will be 3 Del by 3 right it's 3 D by 3 - d s upon 3 and students here we will have + 9 d s upon 9 and here this term will come right leave the rest because we only need power till two rest of the terms are not useful for us right students so when we will simplify this we get it as YN is equals to 1x 3 and 1 minus this will will become D here and this will be D Square by 3 - d sare by 3 and this is D Square from here we will get its value which is 2 D S by 3 okay this will be 2 D sare by 3 and here we will get this as n ^ 2 + n ^ 1 so this is its value right so when you take this inside then YN is equal to 1 by 3 so from here we have two factorial and this will be 1 factorial minus now we will do derivative of this and its derivative will be 2 N right so its derivative will be 2 n + 1 right because if we do its derivative this will be 2 N and this will be 1 and if we do it twice then it will be two so this we will get 4x3 now let's simplify this convert factorial notation again from here we will get its value as n into n minus 1 + n so here we will get its value as n into n minus one when we will do plus n n will be canceled out with n so it will be n Square so I will write it as n Square only as we don't have much space so this will be 2 N and from here minus + 1 + 4 by 3 so therefore it will be + 1 by 3 right okay and this we have is its particular integral this way we can do it here okay let's move on to the next function if we have a product function means a power n into FN then what do we do in that case so if we discuss differential equations there we have fdy if you have studied it earlier we get it like e to ^ x VX so what we do here Y is equal to e ^ x into VX divided by this FD comes here then we will bring out e to the^ X so here where there is D put D + 1 there it means if power is 1 then we will use D + 1 it will be same here we will take out this a to the power and wherever there is e it will be a e here it's plus but here it will multiply then we will calculate its Pi like we calculated here for example we have been given this question here it's given that y n + 1 - 3 YN is equal to n² and 2 ^ n let's say you need to find particular integral here so first what we will do we will write it here so it will be e right and here its value that we will have is YN - 3 YN is equal to n² 2 ^ n this will be e - 3 YN is equal to n² 2 the^ N first we will find CF we know that eus 3 is equal 0 implies we will get the value of e as 3 and the value of CF that we will get it will be C1 3 to the power and right students if we talk about particular integral it will go down we know that YN is equal to this will go below then we will get n² 2 ^ n by eus 3 now we will take this 2 to the^ N Out so this 2 to the power n will come out wherever there is e it will be 2 e remember it gets multiplied right okay this a it will be multiplied by E and this 2 it will be multiplied by 2 e so it will become 2 e right we will use concept of factorial notation so so here it will be y n I told you in the previous question that in factorial notation its value is we have this value divided by here we will write it as D+ I right minus 3 instead of e we can write this y n is equal 2 this will be 2 ^ n this will be 2 factorial and 1 factorial it will be divided by 2 d right and this two will be subtracted from minus 3 so we will get here minus one right students now what should we do we will take out minus common this will be minus 2 to the^ n and then it will go up so 1 - 2 D this will be factorial notation 2 and 1 this will be the value that we will get and this will go up and will be inversed this will be y n is equal to - 2^ n we will open this 1 + 2 D + 4 d s these terms will keep coming as the highest power is two we will consider till that power rest is not relevant now students be attentive multiply this inside when you do this y n will be equal to - 2^ n when you take it inside then this will be N2 + N1 okay we will take its derivative from here we will get its value 2 into 2 this will be 2 N and here we will have its value as 1 we will do its derivative twice so its value will be 2 so 4 into 2 will be 8 we will do its derivative normally like we do simplify this a bit so students we know that this value that came here its value is equal to what the value of n Square I told you in factorial notation its value is N2 factorial N1 factorial so we can replace in square with this this we will get as n² + 4N and what will we get here 10 okay so here this will be the particular integral of this so this way we can solve this question very easily next I will try to explain some more questions like this okay first we'll do this it will be e s YN + 7 e here it will be YN + 10 YN is = 12 and this will be 5 ^ n right what will we do here this will be e ² + 7 e + 10 into YN is equal to this will be 12 here we will have its value as 5 ^ n so what will we do here e ² + 7 e + 10 is equal to 0 right so if we simplify this the value of e will be 5 and 2 okay students so the CF will be YN is equal to C1 5 ^ n + C2 2 ^ n now we'll find the particular integral so why n is equal to 12 5^ n upon here we will have e s + 7 e + 10 okay wherever there is e replace it with 5 this is like the concept of E power x 12 and this will be 5 to the^ n if we will put this it will be 25 + 5 into 7 it's 35 so here we will get 60 and 10 70 here this will be YN is equal to 12 into 5 ^ of n upon 70 so the final answer that we will have is CF plus pi here this will be C1 5 ^ n + C2 2 ^ n plus here it will be 12 into 5 ^ n upon 70 okay so here we have the answer of it and this way we can solve this question very easily so next question is YN + 2 plus YN is equal to sin n < by 2 okay what will we do again we are going to write this so this will be square YN + YN = sin n < / 2 right so here we will get this as e s + this YN is equal to sin and < / 2 right what will we do here we will write e ² + 1 is equal 0 so we will get e is equal to plus minus Iota if we look at its roots then the real part is this and the imaginary part is one we know the value of R will be Alpha square + beta Square so the value of this will be 1 and we will get Theta as tan inverse y by X so beta by Alpha means 1 by 0 so this will be Pi by 2 so the right way to write the CF is this look carefully students r ^ n C1 cos n Theta + C2 sin n Theta clear this will be y Nal to 1 ^ n this will be 1^ n and this will be C1 cos here value of theta is < by 2 so n Pi by 2 + C2 sin n < by 2 so here this will be its CF now we'll talk about its particular integral so what will we get here YN is equal to sin n < / 2 / e ² + 1 where whatever there is a square then its coefficient Pi by 2 we will Square it and then subtract it I have already told you right so YN is equal to sin n < by 2 so we will get this as - < s by 4 + 1 okay when you simplify this we will get the final answer as CF + Pi so YN is equal to we will get its value from here 1 the^ n let's write it and this will be C1 cos n Pi by 2 + C2 sin n Pi by 2 and what will be the particular integral of this 4 will go up right so 4 sin n Pi by 2 divided here it will be 4 - < s right so here we will have its particular integral and its CF right so this is how we solved it here okay so this question is for the comment box how much time did it take you to solve it please comment and tell if you want to watch videos of numerical analysis entire playlist is available here if you are preparing for any competitive exam and want to practice questions whether it is IIT Jam or csir net you can do it and you can subscribe to the channel here thank you very much bye-bye