equation first we will understand the definition of linear differential equation so what kind of od what kind of ordinary differential equation is actually categorized as linear differential equation so let's look into the definition first so linear differential equations are those in which the dependent variable in which the dependent variable dependent variable you can call it as y let's say if you are if you are having dy by dx or d2y by dx square such kind of differential coefficients present in the od then y can be called as a dependent variable and its derivatives occurs only occurs only in first degree occurs only in the first degree now what are the meaning of occurs only in the first degree that means no differential coefficient should have a power more than one on it okay exactly power one should be there whether it is dy by dx whether it is d2y by dx square in fact the dependent variable also should occur in the first degree that means there should be no y square y cube kind of a term okay and one more thing and they should not be multiplied together and they should not be multiplied together not multiplied together so a general linear differential equation would actually look like this let's say I go to a nth order linear differential equation so it will look like this plus some p1 dn minus 1y by dxn minus 1 plus p2 okay dot dot dot till let's say pny equal to some x okay now here please note the coefficients that you see here p1 p2 etc they are all either functions of x or constants including your x1 so p1 p2 p3 etc till pn including your x are either constants or functions of x remember constants can be also termed as functions of x in fact constants can be functions of any variable you want but in this lecture my dear students I am going to only talk about linear differential equations with constant coefficients here so I'll be talking about only those situations in this video I'll be only talking about where your p1 p2 etc are constants so I'm going to talk about linear differential equation with constant coefficients with constant coefficients okay I'll be not going into any other higher version of linear differential equation why I'm mainly interested in LDE with constant coefficient is because you will find that these equations are very much used in the study of electromechanical vibrations and other engineering concepts okay I'm sure you would have all done concepts related to simple harmonic motion right motion of simple pendulum or you would have done oscillation of springs or you would have done oscillatory electrical circuits like lc circuit lcr circuit right lc circuit with emf lcr circuit with emf in engineering we study the deflection of beams whirling of shafts right lot of application is there for linear differential equation my dear in the field of physics and engineering so I'll be only talking about those cases where you have these p1 p2 till pn etc as constants and of course x could be some function of x see a linear differential equation solution is called a complete solution okay which we write it as cs okay complete solution is made up of two things one is called the complementary function okay and the other is called the particular integral other is called as the particular integral so I will tell you how to find both of these components whether complementary function or particular integral for different cases okay and those will be the only cases that will be coming for your problem solving in physics or any engineering exam so cs what we call it is made up of cf and pi okay so complete solution is made up of cf plus pi i so first I will talk about how to find cf how to find cf complementary function before you understand complementary function you have to understand a very small thing which is called the operator d okay so operator d what is operator d normally what do we do is we use some operator to define such operations in calculus for example d by dx is it in like operator d d2 by dx square this is called d square okay d3 dx cube this is called d cube okay and so on I hope you would have got the pattern so if you have the nth derivative happening with respect to x or respect to the independent variable we call it as dn for example d2 y by dx square plus 2 dy by dx minus 3 y this you would write as d2 plus 2d minus 3 y okay now I will tell you why we use this operator d in order to find complementary function we frame something which we call as the auxiliary equation also written as ae okay let me give you an example for example the previous question let's say I want to find out the complementary function for this differential equation okay so what do we do first we write this like d2 plus 2d minus 3 y okay equal to 0 now this equation that you have over here this equation when you create like this this is called the auxiliary equation ae complementary function depends upon the roots of the auxiliary equation how it depends let us look into this example so if there is an auxiliary equation like this we factorize it so let's say the factorization of this will be d plus 3d minus 1 okay from here we get two roots for d okay what are the roots here that we get for d the roots that we get for d are minus 3 and 1 correct now I will not go into the details of how do we get this but the complementary function that we will get will be nothing but c1 e to the power minus 3x plus c2 e to the power 1x where c1 and c2 where c1 and c2 are arbitrary constants are arbitrary constants okay now how to find these arbitrary constants you would be given some initial conditions or boundary conditions into the question through the use of which you can find this c1 and c2 remember don't forget your basics any nth order differential equation will always have n arbitrary constants into it okay now this is a very simple example but let me give you a generic view of this let us say let us say you have got a differential equation which looks like this okay I'm just taking a nth order linear differential equation for you okay let's say this goes till k and y equal to 0 okay so to find complementary solution what would you do you will find the auxiliary equation auxiliary equation would be dn k1 d to the power n minus 1 k2 d to the power n minus 2 up till kn okay put it to 0 put it to 0 now it is not because there was a 0 over here let's say even if there was any function of x you would do the same thing to get the auxiliary equation okay so don't get me wrong here you have not put it 0 because there was a 0 here I have changed it to x x could be some function of x okay so now let us say the following cases will arise let us say this n degree equation in the operator d is factorizable okay and I get these n factors from there okay so when n distinct factors are coming here so let's say it's a case where the roots where m1 m2 etc etc all these values are real and distinct real and distinct okay then in such case your complementary function is given by c1 e to the power m1x c2 e to the power m2x and so on till cn e to the power mnx where again c1 c2 till cn etc are arbitrary constants okay so nth order differential equation will have n arbitrary constants okay don't forget that okay case number two case number two in case number two what if any two of the roots become equal so let us say out of those m1 m2 m3 till mn roots two of them are equal okay let's say m1 and m2 are equal and the other are distinct okay so let's say the others are distinct okay in such case again I will not go into the derivation part because I know you are here to get the idea how to solve a linear differential equation with constant coefficient you don't want to get into too much of theory else this video will become too long in such case your solution will become your complementary function will become c1x plus c2 and the roots which were repeated let's say I write m1 and m2 are same so you can write c to the power m1x and the others will be the same as it was before that is c3 e e to the power m3x c4 e to the power m4x and so on till cn e to the power mnx now a natural question will arise in your mind what if three of them were repeated right so here I took an example of two being repeated let's say if three of them were equal if m1 is equal to m2 is equal to m3 and others were all distinct let's say m4 m5 etc till mn then your complementary function would become c1x square c2x c3 e to the power any one of the name you can take m1x let's say and others would remain the same let's say c4 e to the power m4x c5 e to the power m5x and so on till cn e to the power mnx getting the point so now you would have got an idea that if four of them are equal what to do then there would be a cubic equation coming over here c1x cube c2x square c3x for c4 e to the power m1x then c5 e to the power m5x and so on and so forth okay so you can continue from here right now case number three case number three okay so complementary functions will be required when you are dealing with those kind of circuits where there is no external emf applied or those kind of simple harmonic motions or those kind of spring oscillations where there is no periodic there is no external force applied to the system okay yeah third case is if one of the pair of roots is imaginary now remember we are dealing with k1 k2 as real numbers so remember if imaginary roots will exist they will exist in conjugate pairs so let us say while you are evaluating the root there were two roots which were conjugates of each other alpha plus i beta and alpha minus i beta and the rest roots were like m3 m4 normal your real roots okay so these are all your real roots in such cases what happens your complementary function will become your complementary function will become e to the power alpha x c1 cos beta x remember alpha was the real part of m1 and beta was the imaginary part of m1 okay plus c2 sin beta x and the normal process will work for m3 m4 that is c3 m3 e to the power m3x c4 e to the power m4x and so on till your cn e to the power mnx okay this is the one which is basically used in the simple harmonic motion let's say unforced simple harmonic motion remember you still per harmonic motion used to get your x as a function of t which was having cos and sin term okay so this falls into that category now a natural question will arise in the mind of the students what if what if two of the imaginary roots are equal okay so what if there is a repeated or imaginary root so let us say we have a case where m1 and m2 is your alpha plus i beta and let's say m3 and m4 are alpha minus i beta and rest roots are your normal ones so let's say m5 m6 etc till mn are your real roots coming out from the roots of the auxiliary equation so in such cases again there will be a minor change in the complementary function it will become e to the power alpha x now see what will happen a small change instead of this c1 you will write c1x plus c2 cos beta x and instead of this c2 you would write c3x c3x plus c4 sin beta x and other roots will continue in general like they used to continue in the previous cases so you will have c5 e to the power m5x c6 e to the power m6x da da da da da all the way till c and e to the power mn okay so you will not get more than these cases my dear to find out the complementary function now let us look into few examples i think examples are the best way to understand these concepts let us say i want to find out i want to find out the complementary function for this differential equation okay now thankfully i would take a very simple example here okay so i will take a very simple example here let's find out the complementary function for this okay and let us say i give you some initial conditions that x0 is 0 and let's say the derivative of the function let me write it like this derivative of the function at x equal to 0 is 15 okay now these are your initial conditions since this is a second order differential equation with the dependent variable as x and independent variable st i need two initial conditions or two boundary conditions to get my arbitrary constant okay so let us find out the cf in this case okay now for cf we first need to make we first need to make the auxiliary equation okay for auxiliary equation we get d square plus 5d plus 6 equal to 0 okay this is factorizable okay i get two roots from here which are minus two and minus three okay so your complementary function will become your complementary function will become in this case c1 e to the power minus 2t remember now your independent independent variable is t and c2 e to the power minus 3t okay now luckily in this case your complementary function will be your complete solution because there is a zero on the right side see try to understand my dear students here that particular integral will come into play when there is a function over here okay when there is a function of x over here or some other constants over here right now since there is a zero over here there will be no particular integral so your complete solution would be same as your complementary function okay role of particular integral will come when there is a function of x on the right side of this equal to sign now how do i find c1 and c2 because if i get that i will get my complementary function and hence i will get my complete solution so let us use these boundary conditions x zero is zero means when t is zero the value of x is zero so now remember x will become your c1 e to the power minus 2t c2 e to the power minus 3t so put t equal to zero and put x equal to zero you will get c1 plus c2 that's number one case number one equations and the derivative of it dx by dt if you do you will end up getting minus 2 c1 e to the power minus 2t minus 3 c2 e to the power minus 3t put value of dx by dt as 15 and x and t value as zero you'll get minus 2 c1 minus 3 c2 solve these two simultaneously to get c1 and c2 your c1 value without much waste of time will come out to be 15 and c2 will come out to be minus 15 so in this case your complete solution will become 15 e to the power minus 2t minus 15 e to the power minus 3t so this would be your complete solution for this differential equation again there is no particular integral here because there was a presence of zero i will come to particular integrals when i take such a scenario let us take one more example let us say we want to we want to solve this differential equation again i'm keeping zero on the right side because i don't want to involve particular integral right now we'll talk about it little later on in today's session so if you see the auxiliary equation here auxiliary equation will be d square plus 6t plus 9 equal to 0 here you realize that your different your auxiliary equation will give you repeated roots okay so the roots here are minus 3 minus 3 isn't it it's a repeated root so in such cases your complementary function and hence your complete solution will become c1 plus c2 x now remember there is no x there is a t here so c1 plus c2 t e to the power of minus 3t it will become okay so this will also become your complete solution okay since i have not given you any boundary conditions to find c1 c2 we will leave the answer in this form only okay let me take one more quick example for you let's say i want to solve d3 y by dx cube plus d2 y by dx square plus 4 dy by dx plus 4 y equal to 0 okay again i have purposely taken a zero on this case so that we don't get any particular solution so if you write down the auxiliary equation here what will it be d cube plus d square plus 4d plus 4 okay put it equal to 0 it is not see you're not putting it 0 because there is 0 over here auxiliary equation always is formed by writing the operator d equation equated to 0 okay so this will give you if you factorize this this will give you d square plus 4 times d plus 1 so what is happening here you get one root one root as minus one and you'll get two imaginary roots from here okay so the other roots are 2i and minus 2i correct yeah so in this case your complementary function also the complete solution in this case will become c1 e to the power minus x okay and remember the real part of these two conjugates are zero so you'll have e to the power zero x which you don't need to write actually c1 cos 2x c2 sin 2x i have written it here so that your understanding is clear so ultimately your complete solution will be y equal to c1 e to the power minus x okay sorry i should not use the same name of the arbitrary constants here there will be three arbitrary constants because it's a third order differential equation so the complementary function would be this okay i hope this example was a good enough example for you to understand things all right students let's talk about particular integrals so far we were talking only about finding complementary function now we'll talk about particular integral so particular integral will make sense when you have instead of zero and x over here right so let us say this is your original linear differential equation given to you with a function of x over here okay once you have a function of x over here then only particular integral makes sense remember if there is a zero over here then your complete solution can only be found out by using complementary function so what i want to say is if your x is zero then your complete solution will be just your cf but if your x is some function of x let's say i call it as g of x then of course your complete solution will have a complementary function plus particular integral okay so how do we find out this particular integral let's talk about it so first of all the process is very similar i mean the beginning part of the process is similar to what we use in complementary function first we need to frame up the left hand side of the auxiliary equation which is actually dn k1 d to the power n minus 1 k2 d to the power n minus 2 and so on till let's say k okay so what i'll do is i'll call this guy as f of d okay now in order to find out the particular integral please note you have to do the inverse of this operator on the function x given to you okay so this function x will be given to you in the question you have to do this operation now how do i do this operation and what is 1 by f of d okay so don't worry about it i'll explain you in simple terms let me write down this as let's say f of d is a polynomial in d which is factorizable let's say these are these are the roots of f of d equal to zero okay so let's say i factorize it like this so f of d is factorizable like this so first what do we do is we treat this as if we have a case of a partial fraction scenario like this okay we all have done partial fractions right in class 11th you would have done partial fractions while doing indefinite integrals you would have done partial fractions so let's say you just you break this up into partial fractions like a1 d minus m1 a2 d minus m2 and so on till let's say an d minus mn okay find out your a1 a2 till an by you know your smart substitution method or comparison of coefficient of various powers of d there are so many ways to find out the a1 a2 a3 etc once you have found out let me give you a small formula by which you can compute this okay the formula is here 1 by d minus a of any function is given by e to the power ax integral of x e to the power minus ax dx okay so this is all you need to know okay and you will be able to manage all questions related to particular integrals so please make a note of this now how is this going to help us in solving particular integral see now that you have broken 1 by fd in partial fractions so what are you trying to calculate you're trying to calculate 1 by or inverse of fd on x function okay whatever is your x function that means you're trying to do this a by d minus m1 plus a2 by d minus m2 and so on till an by d minus mn on the x function okay so you can break this up as a1 d minus m1 on x plus a2 d minus m2 on x and so on till an d minus mn on x okay so this is nothing but a1 okay now treat as if you're doing 1 by d minus m1 on x okay so how will you do this very simple we already discussed the formula over here 1 by d minus a on the function x or you can say inverse of d minus a operator on function x will lead into this integral e to the power ax integral x e to the power minus ax dx so in a similar way this fellow it will result into e to the power m1x integral of x times e to the power minus m1x dx okay and same will be true for the others also correct so you will have next one is a2 e to the power m2x integral x e to the power minus m2x okay okay and so on okay so the next term would be a3 e to the power m3x integral capital x whatever that function given to you that depends upon the function so I'm just writing as capital x as of now but you will act as per the question given to you okay so finally the last term also let me conclude here since I've started writing this okay this would be e to the power mnx mnx integral of capital x e to the power minus mnx dx okay right now many times this computation will take a lot of your time so there are certain standard cases that we have already formulated so there are three four cases and those three four cases are always asked in the exam so what we have done we have simplified this act for you through cases okay so I will run you through certain cases by which you can find out particular integral very quickly okay so I'll take case number one first I will start with that case where your function given to you is a exponential function like this okay in such cases your particular integral your particular integral that you're trying to evaluate that means 1 by fd remember fd was the left-hand side of the auxiliary equation okay this term will just become e to the power ax by fa but this is provided this is provided your f of a is non-zero okay so what you have to do is you just have to put this a okay in the left-hand side of the auxiliary equation that you had got so remember you made an auxiliary equation so left-hand side of the auxiliary equation that is what we call as fd the polynomial in d right there you need to put instead of d this value of a but this is to be used only when fa is not zero okay for example let us say let us say I have given you this particular differential equation d2 y by dx square plus 5 dy by dx plus 6 y is equal to e to the power x okay now let us do a complete question on this let's say I asked you to find out the complete solution for this so complete solution will have complementary function plus a particular integral correct now particular integral will feature in over here because there is no more zero on the right side as you can see they have given you a x for that capital X that is the function of x okay so complementary function you already know that you have to form a auxiliary equation look at this and tell me what is the auxiliary equation that you will get for this you'll get d square plus 5d plus 6 equal to 0 correct and its roots are we can all know its roots is minus 2 and minus 3 so complementary function would be c1 e to the power minus 2x c2 e to the power minus 3x so no problem with complementary function now particular integral particular integral will be 1 by fd which is this term remember I'm calling this as fd this whole term here is fd term okay this is your fd term okay so you're trying to evaluate something like this now as this particular method says you just need to put this a in place of d over here so I'm going to put one in place of this will become e to the power x by 1 square plus 5 into 1 plus 6 which is actually e to the power x by 12 so what will be my complete solution my complete solution would be c1 e to the power minus 2x c2 e to the power minus 3x plus ex by 12 now many people ask me sir there is no arbitrary constant in particular integral no there is no arbitrary constant in particular integral please make a note of this a particular integral will not have or does not contain any arbitrary constant does not contain any arbitrary constant please make a note of this very important okay so I hope with this example it is clear that how do we deal with particular integral when your capital X function is an exponential function like this okay so now people say what if sir f a was 0 then how would your formula change so if f a is 0 if f a become 0 please note there will be a small change in the particular integral it would become x into e to the power ax by f dash a okay but this is again provided provided f dash a is not equal to 0 okay again what if f dash a is 0 then what to do then if f dash a is 0 then again not to worry your particular integral let me write it in different color your particular integral will become x square e to the power ax by f double dash a okay and it can go on and on like this so on okay so don't worry even if your f a was becoming 0 or f dash a was becoming 0 or f double dash a was becoming 0 this is the trend which we need to follow now I would not go into the proofs for this because this is a short video I would like all of you to just see in order to deal with your physics problems now case number two case number two I will talk about where your right hand side function is either a sine function you can say sine ax plus b type of a function or it's a cos function cos ax plus b type of a function now here please pay attention it is slightly tricky we need to follow this formula f of d square that means inverse of f d square sine ax plus b is basically sine ax plus b upon f of negative a square okay but this is when your negative f of negative a square is not zero provided f of negative a square is not zero okay similarly 1 by f d square cos ax plus b is given by cos ax plus b upon f of negative a square again the same condition provided f of negative a square is not zero now many people fail to understand this what is the meaning of f d square up till now it was f d how all of a sudden you started talking about f d square and all those things okay let me explain this with a simple example let us say let us say I want to solve I want to solve this differential equation d2 y by dx square let me take slightly more complicated version d3 y by dx cube so that you people will understand you'll you'll think that sir took dx d square because he had a d square in the question okay so let me take d3 y by dx cube plus y is equal to let's say cos of 2x minus 1 I want to solve this differential equation okay so as you know complete solution will have two parts complementary function and particular integral okay for complementary function what do we do let's quickly revise what we had learned we form an auxiliary equation like this okay so what would be the roots of this auxiliary equation minus 1 would be one of the roots okay so let me write down the roots for this your roots would be minus 1 minus omega and minus omega square I hope you all know the complex number minus omega and minus omega square so minus omega is negative of minus half plus i root 3 by 2 and minus omega square is negative of minus half minus i root 3 by 2 okay in short your roots will be half minus i root 3 by 2 and half plus i root 3 by 2 okay now remember for this your complementary function will be c1 e to the power minus x but what happens in case of complex roots you'll have e to the power ax which is going to be to the power half x c1 cos c1 cos beta x plus c2 sin beta x okay remember we had seen this so this is going to be your complementary function now remember this is c2 this is c3 okay i that was a slip of pen this is c2 and this is c3 okay so three arbitrary constants will come now arbitrary constants they will only come in case of complementary function arbitrary constants will never come in case of a particular integral please keep that in your mind very very important okay now let us move on to particular integral so how to find particular integral for this so let me just name a subtopic particular integral so one by f d square now see you have to treat this as a function of d square all of you see how it works so what i will do this d cube i will break it up as a d square into d plus one okay so in place of d square you have to put a minus a square see here the formula this is the formula which we are going to follow here in place of d square you are going to put a minus a square remember a here is two so minus a square here would be minus four so what i'm going to do i'm going to write this like this so one by d square d plus one cos of two x minus one so this will become one by minus four d plus one of cos two x minus one okay now what to do next simple i will i will do one plus four d multiplication both in the numerator and denominator so it was one minus four d so i multiplied and divide with one plus four d okay oh can we do that yes very much we can do that so in the numerator it will remain one plus four d of cos two x minus one remember d is an operator don't treat it like it's a multiplication of two terms denominator you will have one minus 16 d square now again this d square you have to put as a minus four okay so wherever a d square is coming immediately have to put it as a minus four so it will become one plus four d by 65 okay now this operator will work on cos two x minus one now see what is happening on the numerator part all of you please pay attention one into this will remain the same function but when four when d is operating on it it will differentiate it remember d is a differentiation operator okay so the answer that you would get is one by 65 cos two x minus one so basically when one multiplies to this nothing will happen but four d what it will do it will differentiate it so it will give you something like let me put one by 65 outside so four times derivative of this will be minus eight sin two x minus eight sin two x and remember remember derivative four times the derivative of the whole thing derivative here will follow chain rule so minus sin two x into two so eight factor will come over here so this whole thing will actually become your particular integral are you getting my point so this is your complementary function this becomes your particular integral okay so your complete solution complete solution would be let me just copy first what is your particular complementary function c one c one e to the power minus x plus e to the power half x then c 2 cos of root 3 by 2x plus c 3 sin of root 3 by 2x plus one by 65 now I have started writing the complementary sorry I have started writing the particular integral one by 65 cos two x minus one minus eight times sin two x minus one okay this is your complete solution c 1 c 2 c 3 would be arbitrary constant that you need to find out from your boundary conditions now what will happen if your f of minus a square happened to be zero when such things will happen remember the formula will change like this so one by f d square sin or cos whatever you may take it will become x times one by f dash minus a square sin ax plus b now what is the meaning of f dash a square now whatever was your left hand side of the auxiliary equation you differentiate it once correct you differentiate it once and then put the value of d square as minus a square as I discussed with you in the previous problem okay let me take a small example of the same let us say I want to find out the particular integral I want to find out the particular integral for this function only the particular integral I will not do the complementary function part because of the time issues okay let's say I have this question and my question is find the particular integral for this okay so first of all if you see your auxiliary equation left hand side would be d q plus 4d right is it it's basically d times d square plus 4 okay so now now when I do the particular integral formula it is 1 by d d square plus 4 into sin of 2x correct okay now what will happen here if I put remember my a is 2 here so if I put my d square as minus 2 square that means 4 minus 4 will cancel off correct so I cannot put here so what I will do the formula suggests that you need to differentiate this guy so what are the derivative of d q plus 4d with respect to d so you have to do this so that will give you 3d square plus 4 right so now you have to now your formula will change to x by 1 by 3d square plus 4 sin 2x now you can put your d square as a minus 4 so when you do that you end up getting 3 into minus 4 plus 4 sin 2x in other words your particular integral will become x by 8 minus x by 8 sin 2x okay again this is subject to the fact this is subject to the fact I forgotten to write over here this is subject to the fact provided your f dash of minus a square is not 0 okay so a genuine question will arise in your mind what if f dash minus a square was 0 so if that is the case then you can continue the process further that means your particular integral so I'll write it down over here so if your f dash of negative a square was 0 then your particular integral will become x square 1 by f double dash of minus a square sin of ax plus b and this will continue on and on and on till you are getting f dash or f nth derivative of a square as 0 so this is also provided provided f double dash of negative a square is non-zero is this fine okay now we'll come to the last the third case okay third case is where your right hand side function is a polynomial in x let's say I take for general purpose x to the power m okay in such cases what do we do we use the binomial expansion of the given term how do we do that I'll explain this through an example okay this is best understood with an example let us say I want to get the particular integral for this ordinary linear differential equation d2 y by dx square plus dy by dx is equal to x square plus 2x plus 4 let's say I want to solve this okay so in this case what is your left hand side of the auxiliary equation d square plus d right okay now you want to find out the particular integral so particular integral will be what particular integral will be 1 by this of your x capital x this is your capital x term okay so x square plus 2x plus 4 you are applying the inverse of d square plus d see now what you do with this d square by d you'd write it like this okay now let 1 by d be as it is d plus 1 to the power minus 1 okay write the binomial expansion for this okay so it is like I am asking you to write what is the binomial expansion for 1 plus d inverse so you can treat it as 1 plus d inverse if you want for your convenience basically it's it'll be a geometric progression with infinitely big geometric progression with first term as one common ratio as minus d isn't it so it'll be 1 minus d plus d square minus d cube plus d4 dot dot dot up to infinity okay but don't worry you will not require to that extent see when you do d you are doing differentiation okay when somebody says d of x means you are doing d by dx of that term when somebody says 1 by d of x you are using you're doing the integration of that term okay please remember this fact please remember the simple thing it's very very important so what you're doing here you are let's say I keep this operator as of now okay one into this will give you x square plus 2x plus 4 minus d on this will give you the derivative of that so 2x plus 2 will come d square of this will further give you the derivative of this which will be 2 after that even if you do one more derivative it will become 0 so there's no point going further correct now the process is not completed I'll just simplify this so this is going to give you an x square plus 4 correct 1 by d means integration of this so it'll be x cube by 3 plus 4x now many people ask me sir after doing integration can I put a constant of integration no please don't do that I've already told you in particular integral no arbitrary constant will appear so this will become your particular integral in this case got the point okay so these are the three main cases that you will be getting for finding a particular integral okay so I would like to seal this up with a small question all right so let's say we have this differential equation and I want to find out the complete solution for this complete solution means I want complementary function plus particular integral okay now of course since you have something on the right side of this linear differential equation there would be some particular integral coming and purposely I have taken a mix of exponential trigonometric and a polynomial function in this case so that you can you know revise all the concepts so first of all I will go for auxiliary equation here auxiliary equation would be d square minus 4d plus 4 okay put it 0 because that is required for us to find the complementary function so the roots that come out from here is repeated roots from here because this is actually d minus 2 the whole square am I right okay so two repeated roots will come out from here 2 2 so complementary function we can write straight away from here it will be c 1 plus c 2 x e to the power 2 x remember when roots are repeated this is what we follow okay now what about particular integral particular integral will take them one by one okay so we'll break it up as 8 e to the power 2 x 8 sin 2 x and 8 x square so let us first cater to 1 by fd fd will be d minus 2 whole square e to the power 8 e to the power 2 x so 8 will be outside e to the power 2 x now the problem is when you put a 2 here it will become a 0 remember when fd when f a becomes 0 we have to differentiate once and put an x but the problem is even when you differentiate it once more and put d as 2 it will still become a 0 so what we'll do is we'll do 8 x square e to the power 2 x the double derivative of this term double derivative of this will leave you with a 2 correct so this answer will straight away become 8 8 by 2 which is 4 x square e to the power 2 x okay so let me call it as particular integral 1 particular integral 2 now I am calling it as particular integral 1 2 3 because there are three functions there so I'll add them all after we have done with all the three of them so next is 1 by d minus 2 the whole square of 8 sin 2 x right so the other function that I have is 8 sin 2 x okay actually 8 is just a constant that will have no role to play but here also you realize that it will become 1 by d square minus 4 d plus 4 sin 2 x put your d square as minus a square so this will become 8 1 by minus 4 minus 4 d plus 4 sin 2 x okay 4 and 4 will get cancelled off so this will give you minus 2 1 by d of sin 2 x now remember my dear students what did I tell you a little while ago 1 by d means you're doing the integration of this okay 1 by d means you're doing the integration of this are you getting my point okay so when you do the integration of this you'll get negative cos 2 x by 2 correct so this will just leave you with a cos 2 x so this is your particular integral obtained from this trigonometric term right now we'll have to go to the last term which is 8 x square so I'll do that also so 1 by f d that is x d minus 2 whole square of 8 x square so 8 I'll keep outside x square okay as I told you you can use binomial over here so what I'll do first I will take out I will write it as d times okay 1 by or you can write it like this 8 into 1 by d by 2 the whole square correct okay so this will give you 2 and if you take it in the numerator it will become 1 minus d by 2 to the power minus 2 of x square okay let us apply binomial expansion on this so this will become 2 times 1 plus d remember minus 2 into minus d by 2 will give you a d next term would be minus 2 minus 3 by 2 factorial d square by 4 and so on I don't need to go further because if I do it would be of no use because this is a second degree polynomial so d cube will all give you zero zero results right so don't go too far because it will not be of any use to you correct so if I just simplify this if I just simplify this I get 2 times 1 plus d plus 3 by 4 d square dot dot operator operated on x square so this will give me 2 x square so 1 into x square will give you x square d of x square will give you 2x because it will differentiate it d of x square means derivative of x square then 3 by 4 into derivative of this guy which is 2 okay so if I multiply I will get 2x square plus 4x plus plus plus 3 so this is your pi 3 okay so your final pi your final pi would be the sum of all the pi's that you have got so pi 1 pi 2 pi 3 see I've just named it pi 1 pi 2 pi 3 just for your convenience okay so if you don't want to name it like that it is up to you so your final particular integral will be 4x square e to the part 2x then cos 2x and then finally this 2x square plus 4x plus 3 okay so what's your complete solution complete solution is cf plus pi remember cf plus pi what was your cf your cf was c1 plus c2x e to the power of 2x and your pi is this fellow 4x square e to the part 2x cos 2x 2x square plus 4x plus 3 okay so dear all dear all students I hope this video session was useful for you to solve any linear differential equation with constant coefficient it will be very helpful for you when you're solving your simple harmonic motion and you're dealing with oscillatory electrical circuits whether with forced or non-forced functions on it thank you so much for watching


At Audiology Associates of Prestonsburg, you can live your life the way you want and find the freedom of better hearing. You'll experience patient care that is specific to you with exceptional follow-up care that ensures your hearing and balance needs are being met. Audiology Associates at 1428 Northlake Drive in Prestonsburg. Hey size 45th Russell Fork Autumn Fest will take place from Thursday, September 29th until Sunday, October 2nd. This is part of a Virginia Tourism Grant for mountains and music festivals. We have 12 bands lined up over the course of the four-day festival to perform in downtown Hayside. The festival will host games and bounce houses for children as well as craft vendors, food vendors and hay rides. What makes the festival so special is the longevity of it.  The Russell Fork Autumn Fest begins Thursday, September 29th at 6 p.m. For a full schedule of events, you can visit the town of Hayside Facebook page. For Mountain Top News, I'm Kelsey Deem.


I wrote this book, I'm not kidding guys, I wrote this book for me. I wrote it for my young self. If I had this when I was 18, 19, if I had somebody to say, you know what, it's going to be okay. And we didn't get it. We didn't have a hero to look up to in our generation. We didn't have a father figure that said, son, this is how you hold an axe. This is how you work in a car and this is how you treat a woman. We didn't get it. We didn't get that. Our generation has been failed as men. This is why we're scratching our head. This is why we're afraid of the women, because we didn't get that. We didn't get what we've always had in history, which is a rite of passage. We've always had in all of cultures, of all of history, every culture of the world has had a rite of passage from boys to men. The Sioux Native Americans had the vision quest and they would take a young man and a boy and they'd say, go into the wilderness and speak your name. And he'd go into the wilderness, he's naked and scared and he's like cold and afraid. And he'd go out there for days, maybe weeks, and he'd come back after finding his totem animal. This is in a lot of cultures. And then when he came back, they would say, now you can stand with the men. Stand here. The concept, imagine this. The concept of teenager, including the word teenager, did not exist before 1950. The word teenager is invented since 1950. There was no concept of teenager in all of history until now. There was boys and men. And we didn't get that. Our generation has been failed. That's why you're experiencing what you're feeling. Experiencing this loss of connection with women, loss of connection with the female spirit, loss of connection with your masculine edge and your sexual beauty. We've been failed. Our society has let us down. It's not your fault. It's let us down. And so I wrote this book for that little boy who was lost and scared, came out of the wilderness and liked the girls, but was so massively rejected from girls. And tried to be cool and tried to learn things. Tried to be this guy. Tried to get somebody to like me. And I was so insecure because I thought, who would like me? I didn't have an education. I didn't have like, who would want me? Everybody else has got this cool car and had this feeling all these years. And it's like, what you learn is this, gentlemen. And I look at you guys and I see myself. I really do. And here's what I'm going to say to you. There is nobody going to help you but you. You look in the mirror and this is what you do now.   You're not good enough. And that has got to change. That story has got to change. And it's a simple matter of choosing a gentleman. All you have to do is listen. Because your father's not there. They're going to tell you it's going to be okay, son. Here's how you be strong. We have no heroes anymore. Our heroes are 18-year-old rap stars. I want to be like that guy. Who has no clue either. The celebrity culture. So your whole thing from now on, and I mean this very sincerely, is it's up to you to look yourself in the mirror, look yourself in the eye and say, I've got you. It's going to be okay. You have to raise yourself because nobody's going to do it. Nobody's going to do it for you. The Bible says, you know, when I was a child, I spake as a child. When I became a man, I put away childish things. And it's that critical and that deep and that absolute that has to happen. We're a generation of boys who are trying to date girls and we're a tinder swiping. We're a generation of boys who used to play with PlayStation when we were boys. We play PlayStation now, but now we're having fun. We're not having fun anymore. You see? And I'm saying it's strong. And I hope you get it strong and I hope you feel the energy of what I'm saying because it's to me, it's to myself. I wish I would have had a message, but there's no message for men that's real. This is fundamental that goes to the core of who we are as men, which is lost in this day and age.


Okay, so welcome to TypeScript Again, my name is Doris Chen Have a little more detail on this slide So I normally leave my blog address And my email which is doors.chainanimigasum.com Oh, you could always reach me on Twitter at a DorisT-Chain Make sure you put a T there because I don't know who took DorisChain But I got some Comprehensives and I've never replied but I actually I never got it so DorisT-Chain That's my Twitter and so my blog I put all the presentation Technical demo on to my blog, but it is a long URL I actually can't remember that either. So if you do a search Doris space MSDN The probably the top two links one of the link is my blog address So that's a little easy to remember Okay, so I'm a web developer. I have been working on the web space development for many many years and I joined Microsoft about six years ago as a developer evangelist focus on the web space before that I was with some micro system as a Java evangelist for another ten years So I'm really from Java community open source community and really happy to work with you further on that so I like to get a little bit idea about your background How many of you here already have some exposure to Java script? Oh, very good. Awesome. How about a TypeScript? Some of you. Okay. Cool. All right. So let's actually get a Star Wars So first of all How many you never heard about Angular? That's not right. So Angular 2 is actually using TypeScript for their implementation So if you start with working with Angular 2 TypeScript is probably the language that you want to be familiar with and how many you actually using Angular to do development Okay, very good Okay, so we have a pretty busy agenda and I know I have about a you know A half an hour in a 40 minutes to do the talk So I'm gonna keep it kind of like a short but first of all a lot of people always ask me the first questions Why TypeScript, right? So we're gonna talk about that. Why we need a TypeScript? Second thing is what exactly is TypeScript? If you never tried a TypeScript What is your learning curve? How bad it is, right? We talk about that And then what's new in TypeScript because TypeScript 2.0 just released So I really want you to take advantage of all the new features in TypeScript 2.0 And also we're gonna talk about a Common request for me is people usually say hmm. I want to see some demo For TypeScript to work with an Angular app as well as a React app, right? How many of you actually are using React? Okay, very cool. Okay, so let's get in there So first thing first, why TypeScript, right? Because two years ago when I first heard about a TypeScript I actually have the same concern. I was like, hmm, we don't need another scripting language Do we, right? What is that, right? So but the more I look into that the more I fall in love with TypeScript So the idea here is we want developers to be able to focus on creating Amazing things, right? So any developers, if you start any development, you're more likely, what interests you more is Trying to develop some new feature, some new content for the project, rather than Try to understand some other people's code Try to do bug fix or try to maintain the code in a really tedious way, isn't it? So the idea behind is really try to give you more time to work on the things you're most interested Now let's start with something called monocle. How many of you heard about a monocle? It's basically a browser based editor and then it is actually started with JavaScript so at a Microsoft that the whole team is working on and contributing to monocle and And this is actually the foundation of Another tool called visual studio code. How many you heard about visual studio code very good So I'm gonna use it a lot for all my demo and the visual studio code is not just a variable on Windows It also available on Mac and Linux and the most important things. It's a free download. It's free for everybody, right? So we started with monocle implementation. So the monocle project is getting larger and larger So if you look at a typical JavaScript project, you think one southern line is a lot of lines already How about a three thousand five thousand or seven thousand, right? So we see this tendency for monocle project to grow larger and the larger I would say if your project is within one southern lines of code It's still manageable meaning it's probably not too bad to understand it It's probably not too bad to adding some new feature into it But once your project is go beyond one thousand three thousand seven thousand then most of a time You don't even bother to try to understand everything inside a project You would just say if I want to add a feature new feature I will just do a wrapper around the existing thing right and adding something into it so that I don't break the previous code Right, so this is a lot of people just doing it No, even though we know it's not a best practice But just you don't want to break something and introduce more bugs and then spending more time to fix bugs So so because of that we were thinking what it would be a better way of doing things, right? So we did some experiment we take the monocle project as a sample take some statistics And we figure out people spend a 70% of the time just to understand the project itself and Then another 25% to maintain that project only five percent of the time is spending to really create a new feature adding new feature into it So if you look at all the number This is probably not ideal and I would say monocle project is not only one We sort of like it has this kind of number a lot of project larger project also has similar issue So the goal here for us to have another set of Like a super set of JavaScript language is actually this is a rational behind So we want to reduce the time from 70% to 65% to just understand code 20% to maintain the code Give 10% or more on to the creation which is 15% to develop a new code So this is really what we we try to do for monocle project And this is where I show the motivation behind to have type script So type script is really for you to spending more time To develop a new things last time to worry about all the maintenance and understanding so Next thing is what is type script right by definition type script is a type to superset of JavaScript that will comp compiles into plan old JavaScript so basically you're still working with JavaScript But instead of JavaScript we adding more static types onto the existing JavaScript So hence if you go to Type script that you are out. This is a central location if you want to know more about type script It is basically a statically type of superset why it is a statically Typed superset is important because whenever you have statically typed thing You will actually be able to provide the better tooling Right because if you look at a JavaScript, there's another tools available But because it lacks of the static type that feature so a lot of times It's not a really easy for two to support that now we have Type script so it's very much static typed. It's adding the types on top of JavaScript So that's why we could have tools such as visual studio code. We could take advantage of that, right? So the other thing is because of that we actually could scale JavaScript project just like a monocle right right now is you know multiple Thousand lines of code right but with type script we don't feel it's that difficult to maintain and to understand and Again, you know because it's after the comparison it compiled to all the JavaScript. So we'll run everywhere Now so if we take a look the site you could see this is the very first thing for TypeScript and What's good about that is it's completely open sourced. You could actually see the source code on GitHub So if you click on that a particular link you could see exactly What's on get up? So this is all the you know Implementation detail of TypeScript and then you feature anything support you could always take take a look from GitHub Are you all familiar with GitHub? Very good. So that's awesome Okay, so and if you never try the TypeScript if you want to download and install it's very simple Just two lines of command line one is use NPM right to install the TypeScript The other thing is called TSC to compile So these are the sort of like the very first two things you want to know for TypeScript Once you're there you actually could have started with TypeScript development okay, so I Think it's probably better if I could just show you a demo with TypeScript so you could understand much better Okay, so let's move on Visual Studio code. This is a code. All right. So and again, it's free and It also asked me to upgrade so it actually providing those automatically upgrade you could do it all the time so the current version is 2.02 that's my current TypeScript version and What we want to look into is this typical One called the demo one. This is my Particular file and if you're looking to that I don't I don't want to bore you with a detail But this is basically a very simple typical JavaScript implementation Which is I check the weight of particular shipment and then checking the email sending the email sending the updates But one thing I want you to pay attention is if you're looking to this particular JavaScript file In this editor you don't see an error message or warnings at all right So what if I change this can people see on the back because I wasn't quite sure people could see in our back You could actually feel free to sit in the front Framero's are not going to cost more Okay, so I'm going to change this Java script to TypeScript So immediately this will become a TypeScript file and if you look at that right? So when I change to TS instead of JS what I see here is I do have more warning message It does say ah look at the readable So what's what's what's the problem with readable basically? It says it's not assigned to a parameter for this property descriptor So it has a problem with this readable, but remember in the Java script file I don't see an error message isn't it right so with TypeScript it do a pre-checking so we know this particular one is not there Okay, how do we fix it so we could actually go to this define property which is a method and We just go to the definition and you will see this is definition file to defining that a defined property And we'll go through that method you will see there's another attribute called property descriptor So I'm going to look into the definition of this and you will see very easily That's there is a writable parameter there instead of what I put there readable Right, so with this you could actually go back to say ah, this is not a readable this should be Now the little warning sign is gone, right? So that's good. Let's save it The next thing we want to look into is there's another little warning message here. It says the ship wait So what it says it says cannot be applied to type to a number, right? So we were saying hmm ship wait. I thought it's a function, right? It's a method with return, which is here return Something like a number, right? But why it's not a really It's company the reason this company is it forget to put Parenthesis which is a very common mistake in JavaScript board if you don't put if you you think you called a function But if you actually didn't put a parenthesis or whatever it wouldn't come pray it will still say it's okay But with TypeScript, which is more strictly typed it would tell you immediately there's something wrong, right? So you don't have to like typical JavaScript implementation You don't have to wait until a runtime and you see bunch of you probably wouldn't see the error message It because JavaScript you wouldn't be able to see the compilation error message You probably just see the end result either it's a blank screen or you see some error code, right? And then you don't know what's going on. You still have to roll back. Okay, so this at least I know I'm not making the correct a function call so I correct here, right? Okay, so another one which is company here is it says this trim it says this trim should Does not work with a number type so we see what's going on here So let's let's look into this function, which is actually defining this bell and the IDX So for each that a function it's expecting a value and a number Now the way we put it is actually we reverse order, right? So that's why we actually getting the comprehensive value is a number. It cannot do trip. Okay, so that's a reverse it Now those things is actually pretty Common when you're doing JavaScript development because you probably wouldn't be able to figure out exactly what's going on until the runtime But now look at it. It's all gone, right? so very nice, so I save it and I will be able to just to compile it, right? So that's that's pretty good, isn't it now How many of you are actually working with a project which was a multiple JavaScript in the project? Okay, so if that's the case you probably just say, ah, this is just kind of Mickey Mouse kind of file Yeah, you want me to work with but I want to work with a real project. What should I do next? So you look into that I could actually very easy with this visual studio to you could open a command line and You could actually just do TSC dash dash in it so what that what gives you is Default TS configuration file for that Now what's in there? The most important thing you want to pay attention is basically you will say the target of Javascript target for ECMA 6 5 or ECMA 6 some other number, right? But default is ECMA 6 5 Why is that because all the modern browsers more likely will support ECMA 5 But then some browser will support some ECMA 6 not our browser will support that So there will be some kind of distinguish here You will be able to fix like a reset or come something here for your target So that later on you could run in browser, but we will talk about how to work so well with your development cycle So let's let's take a quick look. So this is one thing like I said, this is just one file, right? What if I have multiple multiple javascript So I will be able to use something called allow JS And I'm going to turn this one into true and also I'm going to actually put a little Directory doesn't my output a directory call out So afterwards I'm going to actually save it but before I save it I'm going to do something kind of little tricky here This is something you may want to use. It's called TSC www Dash-dash-w meaning watch What would mean watch mode if there's any change in the project? It will automatically compile to javascript or to all the most updated version right once you save it. So if you're looking to if I save this project configuration file and You should be able to see boom. There is a direct directory called out And there would be a particular javascript called demo.js Which is actually compiled from this demo.ts Right, so that's pretty good and you could actually take a look at the demo.js as actually really Almost identical with the previous TS file. It's just compiled in the runtime to a JS file Now let me actually Open this one to the side So that you could see a little better of what I would do next So next what I'm going to do is I'm actually going to adding some EGMA 6 feature into it Okay, how many of you familiar with EGMA 6? I could define your cross General okay, so this is basically using the EGMA 6 feature Which is I use new definition called a constant constant. I use a lat. I use a cross I create a cross called a customer and I have all the public variables and the constructor, right? So if you don't really understand constant or lat, it's a homework. Go home and figure it out So basically this is the EGMA 6 new feature now Once we put it there remember that the JSON file says the target is five, right? So if I do this and I Do a save and what you will see here is because it's an EGMA 5 So it does not support all the cross definition the constant the lat So you will just translate still as a variable, right? So what does this mean to you mean meaning you could try all the new feature in EGMA 6 or even 7 But then once you define your configuration file the target is EGMA 5 It will automatically compile into the JavaScript which will run fine in all the browsers Right, so you don't have to worry about ah this browser support that feature from EGMA 5 or 6 or 7 You could just directly use it, right? So that's good. And so I will show you something magic So you say what if I change this to EGMA 6? What's gonna happen? Okay? Let's take a look Let's change it to EGMA 6 Let's save it. Ooh What do you see here? So constant lat cross all the constructor all working fine Isn't that cool? So now it's EGMA 6 so you could actually easily because the target is EGMA 6 you could easily, you know Compile your, you know, Java's basically a TypeScript compiler will actually compile all those new feature Into the EGMA 6 compatible JavaScript So you don't have to worry about that you could take advantage doing the all the new feature using all the new feature Let a TypeScript compiler to worry about that, right? So you see I could do Upgrade I could also change it back to 5 to downgrade, right? It's all good, right? So just for the fun, we could change it to Any meter change it back to 5 So very easy to do isn't it? All right So one more thing I want to actually show you here is what if I have a multiple project called another JavaScript called a foo And I want to just leave it a very simple It's across called a foo Commission and Maybe I would do call a method called this spray Me actually go back to the purpose for to make sure it's a calling of the right thing Let me do this first. Let's do it in the constructor adding a spray method Delivery method, sorry, no one gonna find Delivery method, let me call it here and This isn't TypeScript, right? And then let me also call it at foo.js Which is my JavaScript now the reason I'm doing this is because I'm adding allow.js I Will actually adding all that say you could actually change your JavaScript one at a time Say I'm choosing to choose this particular demo one to TypeScript But for the rest of the project you still have 10 say JavaScript file there, right? So you would be able to actually just keep all the JavaScript file leave the way it is and then you could actually See the output which is actually compile all the demo one and the foo into the directory So that you don't have to worry about it changing everything into TypeScript all at once You will be able to do the graduate one file at a time or one function at a time So that will actually really give you some easy time to do your development cycle, isn't it? Okay, now one thing, you know people ask me about Refractor right so how does it work right? How well does it work? Okay, so let's go back to here Let's say I have this delivery method Let me actually close this one because we've done with this and Let's take a look at this guy here, and you will see in the same file. I have two occurrence So I'm gonna actually do Rename into something. I definitely know it's different So you see the very method is changed as well as different method here is changed How about that a full-dollar JS? I just pretty doesn't change anybody want to guess It's all changed to different method So basically meaning refactoring is working across all the JavaScript and TypeScript inside of your project So you could actually change the name convention to change anytime and obviously you would be able to change it from this particular One to the original one. They were remastered to change it in a JavaScript full-dollar JS And if you look at it back everything is changed already in the TypeScript Isn't that cool? So you don't have to worry about if you change something inside your project You have to worry about the change every single file, but your studio code will allow you to do refactoring pretty easy All right enough about this demo. So let's go back to move on to some other cool things so in this demo basically what I did here is I changed a typical JavaScript into a TypeScript and With the TypeScript because it's stackly typed so that you would be able to see Precomparation warning and errors and then you could fix it before you get into the real-time running and Then I also showed how you could actually set up a different target so that you would be able to do downgrade upgrade with a compatible JavaScript of your choice and Then also showed you how to allow a little parameter called allow JS So if you have multiple JavaScript inside of your project You could actually do it one at a time and including all the TypeScript and a JavaScript into the final project Okay, so Afterwards we want to talk about TypeScript, but JavaScript is TypeScript So show me the hands again. How many you actually know JavaScript already? Very good. Congratulations. You could have put a TypeScript on your resume Because basically if you understand the JavaScript, you do understand TypeScript because TypeScript is just a super set of JavaScript Right, but remember I'm a Java. I was a Java evangelist with son before so I do work with a lot of people from Sort of option-oriented background, right? They're not ever really familiar with a scripting language So a lot of times they always ask me they say hey Doris. Do you think I could start with TypeScript? directly without learning JavaScript The answer is no you really have to understand the JavaScript and then you probably 90% there to really leverage a TypeScript. You do have to understand JavaScript, right? It's not a trying to replacing like a Object-oriented kind of press from object rented to to scripting language. Basically it is JavaScript, right? You have to you have to sort of familiar with so this is something basically I would say TypeScript is basically adding optional types Top of JavaScript. So you could look at that. So a lot of times people say hmm I don't care about a TypeScript. Hey, could I still work with JavaScript total fine? You could actually rename all your JavaScript to TypeScript and Without a change in anything or without take advantage of any of the new feature I'm about to talk about in TypeScript. It's totally working fine just by doing the JavaScript to TypeScript itself it will give you all the pre warning message It will help you to develop your code help you to debug your code, isn't it? Okay, so that's what one thing the other thing is you know I keep on getting this question people actually want to understand how TypeScript compiler works And it's very very simple because the idea here is you develop your JavaScript with all the ECMAS 6 or 7 or future Feature and then let a TypeScript compiler to do the work for you You set up your target and then TypeScript compiler will do the work for you So eventually you still work with the the plan or the JavaScript at the end, right? Okay, so the reason a lot of people still have this concern is it is very true if you look at it this kind of like a Web space JavaScript kind of like evolving diagram you will see We actually you know, by the way, how many of you know ECMAS 6 is actually called ES 2015 Very good. Okay, so move on if you ECMAS 7 is called ES 2016 and so okay So this is a new name convention, right? But I know for most developers we still remember ES ECMAS 6, ECMAS 7 instead of put a year there But in reality all the new features coming in every single year because if you look into their name convention That suggests we're gonna have an ES 2017 next year very soon, isn't it? So the JavaScript developer will become more complex I would say a couple of years ago when I talked to people like you guys you guys probably wouldn't I wouldn't ask Say, huh are you a ECMAS 5 developer or ECMAS 6 developer? But now when we talk we probably will say Is that ECMAS 6 you're familiar with or is ECMAS 5 right? So there are more features adding into JavaScript word getting things more complicated And then you have more new stuff to learn and then unfortunately There's always a little gap, right? For example, not all the modern browser support all the features new features from ECMAS 6 and then next year is coming up This is already December, right? So we have a new ECMAS version next year And you will see same story right some browser with support of some features and browser wouldn't so it is actually makes sense to have some Compiler will actually help developers say you go ahead you with a new feature You set up your target and we will worry about the rest of the compatibility issue Who is the browser with the JavaScript version and this is actually a very big thing about a TypeScript compiler All right, but this is just kind of like an insight you don't have to really worry about that because this is already inside a TypeScript So you don't have to worry about how to config TypeScript compile all those things. It's there. It's out of box Okay, but a good advantage is Re-emphasize you could actually take a future ECMAS feature combining with TypeScript and Then let the TypeScript compiler to do the work eventually You will actually compile into JavaScript compatible with ECMAS 5, ECMAS 6, ECMAS 7 or ECMAS of your choice I think the lowest the support is ECMAS 3 All right, so next probably you are sort of Interesting to find out is what's new in TypeScript, right? So I'm gonna actually skip a little bit about the new features because I think just by talking about new features alone It would take another Full session to talk about so I'm gonna skip it just a little bit But then show you these are the new features available for the TypeScript 2.0 as well as ECMAS 6 So the idea of TypeScript 2.0 is fully support all the new ECMAS features We usually take what we call the third iteration of the proposal Once it's complete the third then we start to support right because we know it's probably more likely that feature is going to Make it right so we support it in TypeScript So TypeScript 2.0 is is is a variable and it's very exciting and then there's so many new features into it But I probably don't have room to go over all of them But I want to actually point two of them, which is very very interesting For you to start looking to one is called non notable types. The other is control for all Analysis, I will do a demo to show you what are they but before I do the demo I like to actually talk about what I mean not about types because this is the most Want demanded feature from the community so TypeScript is open source project So you could actually have your voice to be heard You could say I want this particular feature to be implemented in TypeScript And then we look at all the voting right and see how many votes they are will implement that So go back to the notable types if you look at that there are basically Three types of those things number string and a boolean But if you look into different things you will see there is a valid kind of data as well There's also defined and now kind of types within the primitive data type in Javascript, isn't it right? So sometimes you will feel it's it's it's weird because you always have to check if it's not now It's not undefined. I want to make sure it's real data. You always have to do that So in TypeScript we say why don't we remove the other two now or undefined? Outside a primary data type instead of have three primitive data type. I have five right? Number string boolean and then now and undefined right so in that case It's a lot easier to do checking. It's a lot easier to do the rest of the development So if you look at a notable type if we want to represent that number very easy I use a unique type which would say it's a number in this case in TypeScript number meaning all valid data Right or it could be a now data or it could be a undefined data So once we separate these two into such a fashion but use uning to sort of combine them together This actually represents the full blossom data type in the Javascript definition get that all right So what's so cool about that right? That's your next question. Let's actually take a quick look Let me I could I could actually close this project All right, and just leave this project. Okay Are you ready for some kind of interview questions? Okay, this is a very typical question for any Javascript developer you go to interview people will ask you the difference between double ego sign and a triple ego sign Do you guys know the difference? Some people do nodding some people say mm. I have to look into stack overflow This is probably one of the common sort of the most popular topic in stack overflow people still struggle with double triple And I totally agree with you as a as a background is from Java community I I actually don't understand why we have so many things double and triple equals to deal with that I don't know so let's take a look. Can you see it? Okay the code. Okay, so let's take a look So it says this is a testing for Something called s s could be a string or String array or now or undefined so basically says s has all those types data now How do we make sure that a particular knowable type is working in TypeScript when tricking to do is here There is a configuration for there's one called a strict now checks You want to turn it to true if you turn it to force then it doesn't have this kind of knowable new type It will still fall with a Java script So if you don't like this feature totally fine, just turn this parameter into force, right? But now I want to take advantage as I turn it to true. Okay, so go back to this and Would you want to guess if s what is the type of s? Okay, so you have to think about I have so many different type, right? so I Do a checking so this is pretty good with this to it very clearly tell me it's a string or string of array Okay, not too bad and else obviously with all my callable analysis It will say the rest of them is It's a string now and undefined, right? So that's pretty good, right? Now, how about the triple equal sign with object? What is s? As triple equal to object. What is s? You would have seen is the string String array, right? However, did you think there's a now there because now is an object? I Get some of you, isn't it? Okay, so with this tool you could very easily to see there's a now and then obviously the s here is a string and Undefined same thing double equal sign says undefined. This is what this is now and undefined because now is double equal sign with The undefined right so basically, you know I'm not going to repeat of the all of them because basically with the two You would be very easily to see what's going on and then this is a triple equal sign with undefined This is truly it's undefined. That's why this is undefined and the rest of the type is down to the next one Okay, so this is really taking advantage of the normal type and The the the throw analysis think about if you don't have this to you probably was struggling with all the double Equal sign the triple equal sign isn't it you probably not quite sure because you don't know what to expect So this is a really pretty cool feature If you want to take advantage go ahead to do that because I check in my old JavaScript code a lot of times I'm spending is doing the type checking if you go back you probably do the same thing all right I'm almost running out of time, but I actually would love to actually show you a very quick demo Because I actually said in my abstract says it works very well with With Angular as well as with the react which one you want to see with Angular or react All right, cool. I know this is what you're gonna ask. Okay. Very cool. So this is a react Let me actually this is a react project and I guess your question will be because Angular is using TypeScript, so obviously it's easy, but how about a react, right? We have a JS file We have the JSX file, right? So the good news here is TypeScript fully supported react So we actually have JavaScript to TS file. We have JSX to TSX So you could do actually the same thing instead of called JSX it called TSX right so in this particular one It's a very simple Kind of implementation what I did here is basically I'm actually using a Reddit Data which is a JSON data. It's pull all the interesting pictures from Reddit If you want you could take a look what's behind the JSON data. This is the basic the JSON data I'm pulling from this particular site and What I want to do here is I want to actually first of all try to use all the response from a JSON file and then Go through the submission and then And then one thing like what you could see is for example, I said API response What is that I could go to definition which is my another level which is another TSX Which is going to the interface explain what is API response? What is the submission and If you into the index you will see how we set up a URL How we actually go through that and getting all the JSON data We want and then the red is a submission TSX is actually the one is actually getting all the result Display on the page. So let's actually just based on this kill this and Let's do a quick demo. So if I do open command line And if I do HTTP server and you will see EP Logo host 880 You will see bunch of stuff. I'm sorry It's with a URL and with all the time when we submitted the picture But but last night at least is what if I getting an image of there? Let's see how easy it is you're adding this particular image URL as well as the pictures And then you could actually do let's do something quick. I use a web pack power the hosting and Then You will see It should be updated now. Okay, let's do a shift to reload Now you see all the images that in up So this is really using a react take advantage of JSX to TSX and then I probably running out of time But you actually will be able to see quickly the refactoring Everything is is working for example. I could have really refracting everything here To new version as well as I could actually go even to the definition to actually you could quickly show What is really going on there? Okay, so I think I'm running out of time I was going to give you a five minutes for Q&A, but I couldn't stay here Additional time, but if you want to remember one thing, you know, I think this is a slider you probably want to remember this is just go to that site and Take a quick look of this TypeScript and then it will show you exactly what's going on if you're beginner It also has a TypeScript a JavaScript kind of side-by-side kind of tool So you could see how it are going to combine to JavaScript. What's an advantage or new features in TypeScript? Thank you very much


The CEO of a Seattle based company made headlines back in 2015 when he announced that he'd be cutting his own pay in order to boost the Salaries for all of his employees and his name is Dan Price because he doesn't put a price on the worth of his employees But he returned to Fox business news to offer them some advice on what policies would actually Help small businesses and what he recommended It shocked the Fox news host in fact his head nearly exploded You're going to see his response when Dan Price actually recommends single-payer health care specifically to help small businesses This is gold if you want to help small businesses One of the ways to do it is to take the burden of health care away from those small businesses Small businesses like mine could pay a lot more in wages and provide a better incentive for employees Also, if we raise the minimum wage We'll be able to then have that funnel upward with the economy because right now We have a third of all American wealth stuck at the very top with the billionaires If we can have a bottom-up economy that works for everybody we can provide the right incentives for employees So so you think by making it more expensive for a small business. It's going to help them well actually When I started my small business Health care was One of my biggest costs and it still is today But that money actually comes out of my ability to pay employees higher wages So instead of a seventy thousand dollar minimum wage I could have an eighty five thousand dollar minimum wage if we had a health care system that instead of being a giveaway for the insurance Companies actually provided health care for people all over the place. I absolutely applaud Dan Price What he's doing here is so important Fox business They don't like to listen to people unless they're in powerful positions unless they're business leaders So Dan Price is the one CEO who actually goes on Fox News and challenges Their narrative is the only business leader Who actually bursts Fox News's bubble because if you tune into Fox business They're doing nothing but propaganda at the behest of pro corporate America I mean just watch five minutes of Stuart Varney and you'll see how they don't actually even try to appeal to normal viewers This is all about appeasing the business class And it's really transparent. It's despicable. So when you actually challenge their narrative They don't know how to respond and so even after Dan Price very clearly laid out how Making this burden of health care go away with help small businesses The Fox News host didn't know how else to respond So he just says oh so you think that by making it more expensive for small businesses is going to help them No, he literally just said the opposite were you not listening? He said if these small businesses don't have to pay for health care for their employees If this was just something that was guaranteed by the government, they would actually have more money They could actually pay their workers more It's like everything that Dan Price was saying it just went in one ear or not the other and the Fox News Host just had nothing left to say so he just reverted back to his original talking points Oh, well, you want to hurt small businesses with Medicare for all. No dummy I'm literally explaining how this helps small businesses and it's not like Dan Price is some radical socialist who's Saying this whereas, you know other businesses wouldn't necessarily have the same opinion because other small businesses have echoed the same exact sentiment And as Dr. Erica Gonzalez wrote in an op-ed for the Hill the impact the rising costs of health care health expenditures reached 3.8 trillion in 2019 and are expected to rise to 6.2 trillion in seven years are having on Main Street as a serious threat to our economy and Society in fact a national survey of more than 800 small business owners Conducted by small business for America's future found that desperate small business owners are being crushed by health care costs and overwhelmingly demand government action to relieve the burden As an example of how bad the problem is the survey found that 55% of small business owners say the cost of providing health insurance to their employees Is the biggest challenge they face and here's the kicker the difficulty of ensuring workers surpassed taxes corporate competition and even COVID-19 half of survey respondents report providing health insurance as a benefit 76% of the small business owners that do not provide insurance say it's because the cost of doing so is too high and alarming 53% of small business owners who do offer insurance say they have considered dropping it because of the cost not surprising when 9 and 10 survey respondents said their health insurance costs Increased over the past four years with 40% responding that costs have risen by 10% or more a year So it makes sense to suggest that a single-payer health care system would absolutely Be a boost to small businesses in fact some of the candidates who I've spoken with who are running for Congress in Red districts, they actually don't use the usual arguments that I make on this program to sell Medicare for all They sell it as a tax cut for small businesses because it's technically literally that if you alleviate this burden From small businesses then imagine how much more money they'd have Not just to not just to pay their employees, but to actually expand their businesses grow So I think that this is a no-brainer What's shocking to me is just the fact that a CEO said this on Fox business news Like I get that this is Dan Price and he's usually pretty based He's known for saying awesome things like this But you know, it's nice to have one person at least going on Fox news and pushing back against their pro-corporate narrative That's really really awesome to see and whenever I see this happening I always feel the need to showcase it on this program because it gives me hope and when we're in this time where You know the situation the future looks bleak to say the least I think that we need to really celebrate the small victories and I think that Dan Price saying this on Fox news You know Saying this to an audience who wouldn't necessarily be receptive to this message in and of itself That is a victory and I'm gonna take it


What's going on guys? Today I am back with another Destiny 2 top 5 beta video and today we have top 5 luckiest plays of the week and damn some of these are just incredibly lucky they really are. But before we go any further guys if you do enjoy the video leaving a like is much appreciated on my behalf. Can we reach 3000 likes? That'd be absolutely epic if we could. Okay so getting straight into this top 5. In at number 5 we have a clip centred by Jordan Bust 2001 who starts off this top 5 with his great triple.  We have a clip centred by Contour the Joel. Okay so did he just get a throwing knife double kill. What is this? In at number 2 we have a clip centred by Grim Reefa 0420 the second of this kind of scene still as awesome as ever. Oh triple club. In at number 1 we have a clip centred by I'm NLB Pro. Okay so this is going to be the top clip for a while. Watch and you will understand. But guys that is the end of this top 5 luckiest players of the week. If you enjoyed the video leaving a like is much appreciated on my behalf. If you have anything bad ass to send in send it into the email I've just linked at the top of the video description and as always guys see you on that next one.


The proposed methodology is based on the use of the Lambert W function to transform the series resistance equation into an explicit expression. It can be used to analyze the performance of commercial solar panels under various levels of irradiation and temperature. The results obtained are consistent with those obtained by other methods, but they require less data and can be easily implemented by small scale users. This article was authored by Javier Cubos, Santiago Pondado and Carlos de Manuel.