All right, play it safe. We can still make this comeback. No, we can't. Yes, we can. No, we can't. How's it going everybody? Datsunoi here back with some more dragon ball fighters, rank matches. This time playing a variation of the main team with Krillin, UI Goku and Yamcha. Hey, the Genemba intro. Who could forget? And he's gone. Well, the new version of Genemba's here. Oh, I always forget that it actually does keep the in-game color. Oh, back it up. Oh, another UI Goku player, huh? Dragon rush early on in the set. We got to set the tone. Oh, man, nothing like an early dragon rush to make both people in the game realize that this is a, this is not going to be your standard fighters affair. If you, if you do an early dragon rush, it's just the tone of the game gets very different. I'm just going to run those rocks again. I love this team. That was sick to age. Down heavy, just keep it in vulnerable. I'm at level three. I have to level three. Okay, right now I've done literally everything you can do to make this guy never want to click a dangerous button against me. That was nice, actually. I think that was sick. The read on me doing something fake there and him just jabbing was great. Genemba's auto combo actually pretty good, by the way. It's not a UI Goku or Bardock level, but it does have some Bardock tendons. That was almost good. It does have some Bardock like characteristics, specifically the second hit. Can we maximize this? I don't know. All right, we're good. I'm actually going to level three here. We kind of messed up on our chance to get Genemba's blue health, but as long as he doesn't UI tech, please don't UI tech. I didn't want to commit to anything past that. I might spark. You already use your spark. Oh, that was bad. I think it would become a bad decision so quickly. That was actually insane. There it is. The UI second hit. That's it. Now that's a second L, dude. UI's auto combo. That's second L. That's a beast. Destroy me with a level one. Probably going to switch out here, which is the worst case scenario for us. Oh, end to a level three with Yamcha. That's bad. There's no other way to say it. I think he might go cross up. Oh, dragon rush. Fair play. I regret, by the way, I regret using that sparking so bad. I regret it as soon as he hit me. But now, now it's a whole other ball game, dude. Now I really regret it. We need to get Yamcha either out of here or just, oh, okay. That works too. As much damage as possible on Yamcha. All right. I'm going to try to do it through Krillin. I'm going to try to do it through running the rocks again. This time we go low. He's not giving us a lot of time. I say we hit him with just the right amount of stagger now. He's not actually got hit with the overhead. Never mind. Go for the kill. We're in a 2v2 situation now. We're in a 2v2 situation. I think, honestly, I think I'm more scared of Yamcha. I don't know why. That last time, the beam. That was sick. And because Krillin actually has some really good post vanish, we're right back in the corner. Run the rocks. I messed up. I messed up. I didn't combo off of that. That was huge. Oh no, we have to guess again. I mean, it's not really a guess. You could just play defense, but I'm so bad at that part. Oh, I didn't block. That's actually the easier one to guard, too. You just hold down back. It's fine. Uigoku can get this done. Oh, come on. I trusted your Uigoku move. The flip. I don't know what it's called. I'm going for it. Oh, DP, fully invincible. I can't believe it went off. And now it's really clutch time. Next hit wins, I think. Whoa, what are you doing? Oh, no, no, no, no, no. Wait, wait, hopefully not next, next hit. He's got two bars, though. He's got two bars. This is bad. This is so bad. He didn't do it. Nice. I can't believe he risked it all on the counter right there.  In fact, I thought about doing the counter after we missed the UI tech, but I'm glad the DP option also worked out. Why was game one so... Game one did not need to be that intense.  Oh my, oh my God. That was a very aggressive standpoint. I don't think I've ever seen that one.  Then it was Yamcha spirit ball. I could appreciate that at least. I could appreciate the Yamcha aspect. Ah, the Key Blast is running the game. Oh, he just tried to teleport on me. Really, not, not... There we go. Okay. I was going to say, he's been hitting me. I think his reaction time might just be really good because he's been hitting me out of the fake stuff I've been going for. That was also really... Yeah, I think he's just got... I think we got a case of somebody with good reaction time. I'm doing it. I don't care. I'm risking it all. I regret it. Why do I keep regretting my sparks? Woo! That was kind of impressive though. I'll give it up. Just counter, just counter, just counter. No, I can't believe we messed up three times. Oh, we missed the perfect counter opportunity. Oh, again with the luck. Oh, it didn't work. We're back in this. I can't... Okay. Dude, I don't know what's going on in this game. I've been wrong three times already. Nice. Now it's our turn to Yamcha level three. Now he's got to play the guessing game. Do you A block low, B block overhead, or C block cross up? A. I didn't purposefully didn't give him all the answers to the test until I decided which one I wanted. Get him out of here. All right, our Yamcha won this matchup. Ironically though, this time he's the one that still has sparking. Yamcha. Okay, I'll take it. Not the outcome I wanted. Rock! You got to respect the rock. I don't really have anything else I could do here. Backflip. Oh, no. I didn't think I'd connect. Misconnection goes a long way into making sure that I think we're going to take a level three here. No, no. He doesn't have Yamcha. We'll probably just... Oh, wow. We just let us go. Nice defense. No command grabs allowed. Nice. All we have to do. The raid boss is incoming though. Uaigoku with spark. Obviously he's pretty low on health. He's probably one touch. But that doesn't mean he's going to make the one touch easy to get. This is Uaigoku we're talking about here. Can he kill here? Oh, he's... She really went for it. He's trying. I don't know if it's enough. It might be. Oh, please tell me it's not. It's not.  Yo, who said that, bro? Who would yell at Uaigoku like, that's crazy. I don't know which one of you said that, but you're kind of wild for that. 2s. Nice. You know what? If I'm going to beat Uaigoku, I got to beat Uaigoku, baby. If I was sick, come here. Flip him around the back. Come here, man. We'll take two games up and we'll be getting out of here. I think, right, we won the first game. I'm pretty sure. Man, what a first set to start off with. That was intense. Ooh, 3,000 points too. Almost 4,000. We are moving on up. Well, this is a rare thing to have happen in one of these ranked match videos. I mean, it happens sometimes, but we're actually getting a rematch. He run it back. We met each other again and ranked. Arxis, I'd love for the system to go back. It's actually so much fun running these longer sets, like three to four games. They don't have to revert it back to unlimited. I don't think so anyway. I mean, I'd like it if they did, but you guys can let me know down below in the comments. If you get to this part of the video and have an opinion on that, let me... Oh, no, hold on, hold on. Let me know. We're getting Genemba back. Obviously, we got to take Genemba back. We got to go. All right, but honestly, I think if they just did like four matches or three, instead of just the first to two, I think that'd be better. Way better in my opinion. I think this is a game that's decided on three matches. Now, hold on, back to the game. We got a great start. 2S, no, he didn't feel that. Oh, Yamcha assist. I see why people get hit by it now. I definitely went to Solar Flare. Please don't level three with Yamcha. I appreciate it. Defense, try to focus. Don't let anything break our guard. It broke my guard. That was sick. It created so much space. Nicely beat. Wait a minute. Wait, don't kill. No, I need these Sensubines. Don't do this. You don't have to do this. I need them. Wait. Oh, I thought he might have dropped it. Wait. Oh, no, that caught me. Never. I told you guys to wait a lot there for no reason. He actually got my Sensubine machine. Dude, I was going to heal up my team with those. That's fine. We'll probably be all right. That was fast. Well, I couldn't get unstuck, unfortunately. Instead, we're just going to have to UI Goku it out. Launch him up. Do we level three here? Nah. Wow, that recovery was bad. Stop that beam right now. I'm just going to spend it all. Why not? Do I actually spend all of the meter? Nah, we'll just take a level three. I was going to throw out a level one, but I feel like it wouldn't kill. It's just so unnecessary. I'm in a backflip. Backflip again. Oh, that didn't work. Okay. No spark from him. He wants to save for his last character. Grab. Level one probably won't be enough. Just barely. And Yamcha took a beating from that as well. I think we instantly switch out to UI Goku here. That's it. That's game. UI Goku. Why is he not dying? Oh my goodness. He was really hanging on. We were not doing any damage to him there. It's fine. In the end, we did get the win, which I'm really glad about. I'm glad I got to fight side by side with you again, Krillin. Rematch. Let's see if we can't finish this off. That was awesome. I wonder if he has any more. I will never fight again. Wow. All of his stamps are so aggressively, are so aggressive. They always jump to the farthest conclusion.  I'll never fight again. Oh, lag drop. No, bad reflect for me as well. That was kind of sick. I don't think that was it. I don't think that was a part of the play, but I think it was the backup strat. I'm just getting styled on right now. Hold on. I can't take dragon rushes. I don't know who I thought I was. I really thought I was taking dragon rushes today. That's crazy. Oh, bad start for me. Actually, I feel like Krillin's getting destroyed in these matches. Come here. Overhead actually caught me. I didn't have time to do the DP. Wait, I need you to stop. I need another chance. Do I UI tech you? Wow. Level three. I did not expect that. I don't think we do it. It's too obvious. Yeah, it was way too obvious. It's cheap. Get back into the corner. Can I introduce you guys? I'm going to level three here for the blue life, for the blue life and the mix up. We're going to go low. He's got a block low cross up EX will feng fist. I'm telling you the exact strat. I told you in advance. Looks like somebody didn't get the message in a bottle. We take those. Oh, it's not going to be enough to kill. Why did I do that? And I missed it. I'm probably going to be a little quiet. I just I have to grieve. Oh, wow. That connected as well. Okay. We are looking. We are going to need a massive comeback here. Even if he's probably going to level three.  My sparking timer, please. Whoa. He tried to hit me with the genema mix. I don't think so. Whiff punish. Is this really Dragon Ball? I just whiff punished that. That was crazy. That's not good. That's really not good. Oh, man. So much damage. Three bars. He could level three with Yamcha here. I think that would be a pretty decent idea. Did he just did he just? Oh, I can't believe we're alive. Oh, he crossed me up at the last second. I felt too slow. Did he just jump in and then down heavy in the air? Did he hit me overhead?  All right. Play it safe. We can still make this comeback. No, we can't. Yes, we can. No, we can't. I welcome aboard the roller coaster that is my state of mind. Hold on. Oh, wow. Wow. Now that I'm looking around at my house, I can actually kind of make out the circus tent shape. I just never realized I've been a clown this whole time. That's crazy. Okay, please. We can't let it wrap up like that, though. We can't let it wrap up like that. We got to finish stronger than that. I need a win or at least a respectable loss. Please. Do I solar flare? Round start? It didn't work. It's fine. Yamcha catch. Oh, I mean, okay, we still got it, but I played that pretty poorly. Now we are going to run those rocks. So we take the normal elves, run the rocks. Nice defense. Nice defense. Oh, man. Unfortunate. Got the teleport combo. Vanishing bet. Putting me back in the corner. What, what is going on? All right, fine. If you want the corner, bro, you can have it. I was, I was never that sentimental about it. Thanks for the offer, but I actually don't want the corner anymore. All right, we're not going to level three here, mainly because we don't have three bars, but I got you the whole time. Anime. Oh, no longer anime. All right. Well, anime's been canceled, boys. We had a good season and a half. I think I'm going to UI tech here, honestly. Switching out is pretty smart. That kind of changes things. I don't think I should UI tech anymore, but I will. I will. No, so close. DP also hit me. We could really use a sensei bean right now. Whether or not I'll get one, I don't know. Oh, no, we're not, we're not getting one. Yeah. I realized I didn't want to spark and I also was bad at defense. So that was probably it was looking like it was lights out for me. Oh, I switched. I switched. That's a lot of blue life. He probably has to switch me back out, which I would understand. I wish I would have gotten that sensei. Oh man, I don't like it here. Okay. Okay, we got to switch out, right? No, forget that, dude. I'm going in. I wish I would have switched out. Nevermind. Nevermind, I'm fine. I'm cool with this. EX flip. No. Okay, we're in a bad place. We're in a bad place right now. We've made a lot of mistakes. I wish we would have gotten a sensei out to Goku. I can't believe we didn't get hit there. Honestly. Okay. 6M was rough. Oh, we got caught in the scramble. You know what? I'm going to put all my faith in Yamcha. I got to put all my faith in Yamcha at this point. The only character left standing between me and crushing, crushing defeat. I got to focus. All of a sudden, everything's on the line. But when you believe in your characters, bro, things change. Boom. Let me get both of them. Two for one special. Bogo. Hold on. I'm still believing.  Spirit ball. Let's go. And now it's the Yamcha fight. I hope he does what I'm doing. You didn't sign the Yamcha pact of agreement, dude. You didn't read the terms of service. When you play Yamcha, you go, he messed up the level three slightly due to lag. I am the Wi-Fi warrior. I actually feel so bad about finishing that game that way. Oh, see, this is what arx is. You've got to think, I can't leave like this, but it's going to force me to. Hopefully you guys enjoyed this video. I'm so sorry to the person that I was playing here. God, that was so unfortunate. Hopefully you enjoyed it, though. I appreciate you guys watching the games, and I had a great time. Great games to my opponent. While you're down in the comments, let me know your favorite one. Make sure to hit the like button and subscribe to the channel if you want to support and catch even more videos like this. Another way to watch even more videos is to click on those that are on your screen. I have been Datsudoya. Thank you so much for watching this one all the way to the end, and I will see you in the next one.


Summer is here and the 12 mini deal just got even hotter. The iPhone 12 mini is now one penny with a two-year contract. Visit an Appalachian Wireless Retail location or go to AppalachianWireless.com to view this amazing deal. We are you. We are Appalachian Wireless. The Leva Amp Music Series continued in White'sburg, Kentucky Thursday evening with none other than blues rock legend Eric Gales. We're having a great time. The crowd is enthusiastic. The band is amazing. Eric Gales is possibly the best guitar player in the world according to Carlos Santana and Eric Clapton and Joe Bonimosa. Three guys who ought to know a little bit about guitar playing. So far, the series has consisted of several national touring acts such as the Rad Trads, the Ted Yoder Group, and the Kentucky Head Hunters with more acts coming up every Thursday. We're really happy to be able to bring great acts in. We really try to zero in on high-caliber musicianship and we're definitely getting that tonight with Eric Gales. During each music series event, several vendors set up such as the farmers market and food vendors so attendees can enjoy food and drinks while watching a spectacular live show. But Ruth says the concert series has more than just live music. It's about health and wellness. Appalachian Regional Health Care is our presenting sponsor. Mountain Comprehensive Health Corporation is one of our platinum sponsors. And it's about wellness. They believe and we believe that bringing live music into the town will make people feel better, make people happy. And it's all about living well and trying to set up a great environment for our local folks to live in. For more information, visit grant.levemp.org. So we've got Susie Boggs coming up next week on the 19th. We have Arlo McKinley coming on September the 16th. Magnolia Boulevard shuts things down on the 23rd. We have Grammy winner Terence Simeon and the Zydeco Travelers coming on September the 9th. So we've got quite a few shows left, all high-caliber, high-quality musicians coming to Weizberg. So everybody come out. It's free. It's family-friendly. There's no reason not to come at this point. If you can go see world-class high-caliber music for free with your whole family and maybe even getting fed at the same time, I can't come up with a good reason why you shouldn't come. We want people to be safe. We want people to come out. We're having a good time in the outdoors. So please come on out. It's a great time here in Weizberg. Reporting for Mountain Top News, I'm Jeremy Justice.


Good morning. As we saw in the last lecture, the method of undetermined coefficients for solving the linear non-homogeneous ODEs is limited by the necessity of constant coefficients and it can handle only a few selected functions as the right hand side, namely the polynomials, exponential function, sinusoid and their combinations. Now, for general situations in which the right hand side function can be anything, the input function can be anything and the coefficients are also not required to be constant if they are functions, if they are variable coefficients, then the general method for solving the non-homogeneous differential equation is the method of variation of parameters. In this, we first note that if y 1 and y 2 are two linearly independent solutions of the corresponding homogeneous equations, then this combination, this linear combination with constant parameters C 1 and C 2 gives us the general solution of the homogeneous equation, that is the complementary function for the purpose of the non-homogeneous equation. Now, for the non-homogeneous equation, the solution needs to be linearly independent of these two basis members. Now, as long as these parameters are constant, we cannot have linearly independent function. So, what we try to think is that how about making these parameters as variables, that is as functions of x and as we do that and we replace C 1 and C 2 by two functions u 1 of x and u 2 of x, then what we are doing? We are conducting a variation of parameters and that is why this method is called variation of parameters. So, we consider variable parameters.  Now, one point is very clear that we can take this y p and force it to satisfy the differential equation, that will mean that we will differentiate it twice and then take this y p and the resulting y p prime and y p double prime and insert in that equation and we will say that this equation is now satisfied, but that will give us a single second order ODE in u 1 and u 2. Now, there are two unknown functions to be determined and for that a single differential equation is not enough. We will need two differential equations to solve for two unknown functions. So, one more condition is needed to fix them. Now, at this stage we need one more condition between u 1 and u 2 apart from the condition that together they have to make y p satisfy the differential equation. Apart from that we need one more condition and we try to frame that additional condition in such a manner that our work is also reduced, our work is also made more easy. The way to do that is at first as we differentiate y p this function this proposed function we get u prime u 1 prime y 1 plus u 1 y 1 prime from the product rule.  So, these are the four terms in the derivative of these two terms here. Now, we know that as we differentiate it further then u 1 u 2 y 1 y 2 all four of them will generate their second derivatives and that means that the resulting expression for y p double prime will have up to second derivatives of all the four functions y 1 y 2 u 1 u 2. With y 1 and y 2 there is no difficulty because they are known functions but, u 1 and u 2 when they are when involve their second derivatives that mean that the resulting differential equation will be a second order differential equation. The additional condition that we need we can choose in a particular manner which will obviate that second order. So, what we do is that from here we collect the terms which has the first derivatives of the unknown functions u 1 and u 2. So, that means this term and this term and together we say that we will impose the condition on functions u 1 and u 2 that this term and this term together vanishes that is we impose this plus this equal to 0. If we do that then y p prime reduces to the sum of the rest of the two terms this. Now, in this expression in y p prime we have got the derivatives of y 1 and y 2 but, not the derivatives of the unknown functions. So, that way when we now differentiate this we get the derivatives of the unknown functions only up to first order. So, that is this y p double prime that again with the ordinary product rule will give us u 1 prime y 1 prime plus u 1 y 1 double prime and similarly two terms from here right. Now, here again we find that the unknown functions u 1 and u 2 are involved only up to the first derivative known functions y 1 and y 2 are involved up to second derivatives but, that is not a difficulty. Now, on this note that another condition like this we cannot impose because if we impose another condition like this that is if we try to say that these two together also vanish then that will not make sense because this condition and that condition together will imply that u 1 prime and u 2 prime are all are both 0. Because these two this condition and this condition together will mean that we are looking for u 1 prime u 2 prime which satisfy a homogeneous system of linear equations with non-singular coefficient matrix. So, that will give only the trivial 0 solution. So, u 1 prime u 2 prime will become 0 which means that u 1 and u 2 will turn out to be constant and that will not give us the solution of the non-homogeneous equation. So, obviously only one condition we needed and that we have got the second condition we need to get by substituting this y p double prime this y p prime and this y p sorry this y p double prime this y p prime and this y p into the differential equation. As we do that y p double prime these four terms plus p into y p prime plus q into y p is equal to r x this is the given differential equation. Now, as we rearrange these terms collecting together all terms which have u 1 that is this is the first term then from here we will get p y 1 prime and then from here we will get u y 1. So, we collect the u 1 terms together and u 2 terms together. Now, we know that y 1 and y 2 are solutions of the corresponding homogeneous equation that means that this equal to 0 and similarly this equal to 0. So, then considering these two 0 terms we get u 1 prime y 1 prime plus u 2 prime y 2 prime as equal to r x. Now, this condition that we imposed through which all we also simplified the expression of y p prime through which we avoided the second derivatives of u 1 u 2 in the resulting equation here that is one equation and the given differential equation gives us the second equation in two unknowns are u 1 prime and u 2 prime. Now, this equation and this equation if we write in the proper matrix vector form then y 1 y 2 y 1 prime y 2 prime come in the coefficients and we get the system of equation in the standard form. Now, if we solve this then we will get u 1 prime and u 2 prime and we already know that this matrix is non singular for all values of x under question in the interval for which we are looking for the solution. Now, the determinant of this matrix is actually the Ronskian. So, as long as y 1 and y 2 are linearly independent solutions forming the basis for the solutions of the homogeneous equation. So, we know that these are two linearly independent functions. So, this Ronskian will be non zero. So, that means that the solution of this will give us unique u 1 prime u 2 prime in other words this is an invertible matrix. So, we get that solution as we get that solution from the two equations here we get u 1 prime as this and u 2 prime as this Ronskian goes in the denominator which is not a problem because it is non zero and then this is a known function of x. Similarly, this is another known function of x. So, direct integration will give us u 1 and u 2. These u 1 and u 2 when put in the proposed form of the particular solution will give us the particular solution that we are looking for that is u 1 y 1 plus u 2 y 2. So, this is the method of variation of parameters to find the solution of the non homogeneous differential equation in a general situation. And the first step even in this is to find the solution complete solution of the corresponding homogeneous equation that is the first step because they will form this matrix here. So, this is applicable for all functions p x p q x and r x which are continuous and bounded and it will be valid only for that interval in which these functions remain. So, in this lesson these are the important points that we have studied after looking at the function space perspective of linear odys we studied these two methods undetermined coefficients and variation of parameters. These same methods and the previous discussions on finding the solution of the corresponding homogeneous equation all these now we generalize for higher order differential equations. So, first we consider this equation this is the general linear or a differential equation of say n th order. So, up to the n th derivative is appearing linear and these are coefficient functions which depend on x only. Now, understand again that we are here going to consider only those cases where p 1 p 2 p 3 up to p n and r these are functions of x which are continuous and bounded in the interval under discussion. Now again for this we will have the general solution in this manner as y h plus y p where y h is the complementary function in other words y h is the general solution of the corresponding homogeneous equation that is this equal to 0 that is it is a solution it is a it is a general solution of this equation. Now for the homogeneous equation suppose we have got n solutions y 1 y 2 y 3 etcetera then the state vectors of these solutions when assembled as column vectors in a matrix then what we have it is an n th order differential equation. So, state vector will have the function itself it is first derivative second derivative up to n minus 1 as a derivative. So, this is the state vector of the first solution. Similarly, this is the second vector second state vector for this corresponding to this and so on. So, when you assemble these as columns then we get an n by n matrix of functions that is the first row is these functions second row will have its first derivative third row will have the second derivative fourth row will have the third derivative and so on the last row the nth row will have the n minus 1 as a derivative of these functions. So, this is the fundamental matrix which will appear again and again now first point is what is the Ronskian in this case. So, the determinant of this matrix is the Ronskian in this case this is a direct generalization of the case of second order differential equation. So, this is the Ronskian of all these solutions all these n solutions. Now parallel to the theory that we studied in the case of second order differential equations here we will have corresponding points. First point is that if these n solutions of the homogeneous equation are linearly dependent then the Ronskian will be 0. How do we prove that? We say that if these are linearly dependent then for a non-zero vector k we can find this that is we can have a linear combination of the n solutions 0 even though each contribution is not individually 0 that is for a set of k's k 1 k 2 k 3 up to k n not all 0 together we can find a linear combination of these y's which turns out to be 0 if they are linearly dependent. And if so that will mean that the derivative of this will give us sigma that is sum of k i y i prime as 0 is k i y i prime that is j is equal to 1. The derivative of that will give us k i y i double prime equal to 0 that is for j equal to 2 and so on. That means up to n minus 1 we can construct these sums and get them 0 if the solutions are linearly dependent. Now note that this will mean this and this together will mean that the fundamental matrix y into k is equal to 0 which means that its first row is this, second row is this for j equal to 1, third row of this matrix vector equation is this same thing for j equal to 2 and so on. So, the complete n rows we can construct like this and note again that if y 1 to y n are n linearly dependent solutions then we can construct such a matrix vector equation homogenous equation 0 on this side and have a solution which is not 0 that is vector k is not 0 and that means that this coefficient matrix y must be singular because for the homogenous system of equation to have a non-trivial solution k it is necessary that the coefficient matrix is singular. So, y axis is singular which means that it is determinant the Ronskian is 0. So, this is the first point that is the n solutions of the homogenous equation being linearly dependent will imply that the Ronskian is 0. Now if the Ronskian now we try to show the converse if the Ronskian is 0 at some point then y this matrix at that point evaluated at that point is singular and then that will mean that a non-zero vector k can be found in its null space which will in turn give this which will imply that the solutions are linearly dependent. So, this is the converse of this now together they also mean that 0 Ronskian at some point x 0 will immediately imply 0 Ronskian everywhere because 0 Ronskian at some point will mean that the solutions are linearly dependent which in turn will mean that the Ronskian is 0 everywhere this x and that will again mean that non-zero Ronskian at some point will ensure non-zero Ronskian everywhere and the corresponding solutions as linearly independent. So, now what we do with those linearly independent solutions if the Ronskian is non-zero we say that with these n linearly independent solutions of the homogenous equation we have the general solution in this manner as we had for the case of second order differential equation and this linear combination the complete solution of the homogenous equation will serve as the complementary function for the solution of the non-homogenous equation. Now, we consider the first case of constant coefficients consider this constant coefficient case for homogenous equation for which the complete solution of homogenous equation we can find if the variable if the coefficients are functions q 1 x p 2 x and so on then in general we are not able there are no analytical procedures to find out all solutions of this homogenous equation. Now, in this case if the coefficients are constant then we can do that exactly as the second order case with necessary extensions and changes here also the left hand side of this equation has y y prime y double prime in linear combination. So, we expect solutions of this form exponential solution. So, if we take a trial solution of this kind and insert it its derivatives as many times we differentiate that many lambda get multiplied with it. So, we get the auxiliary equation as lambda to the power n plus a 1 lambda to the power n minus 1 and so on till from here we will get a n. Now, this is a polynomial equation of n th degree that means it will have n solutions. So, out of from those n solutions for every simple real root that is if gamma is one solution of this polynomial equation appearing only once that is in its factorization if lambda minus gamma appears only once that is that makes it a simple root. So, then corresponding to that e to the power gamma x will be a solution that is for every simple real root the corresponding exponential function will be a solution of this differential equation because that is the idea with which this was started for every pair of complex roots like this mu plus minus i omega. You will say that e to the power mu plus minus i omega will be two solutions in d d s and in the case of second order differential equation we reorganize it and found the two solutions in a different manner that was e to the power mu x into cosine omega x and e to the power mu x into sine omega x. So, these two will be the corresponding to linearly independent solutions of this homogeneous equation. Now, if we get a real root which is repeated then if it is multiplicity is r then we will find that e to the power gamma x x into e to the power gamma x x square into e to the power gamma x and so on up to x to the power r minus 1 into e to the power gamma x all these will give us r linearly independent solutions of this. This is something which you can prove which you can show similarly for every pair of complex roots which appears repeated you will have similar additional linearly independent solutions. For example, if mu plus if mu plus minus i omega appears twice. So, that should account for four linearly independent solutions. So, first pair is e to the power mu x cosine omega x e to the power mu x sine omega x and the next pair will be these two multiplied with x. Similarly, if it appears twice the third pair will appear with x square e to the power mu x cosine omega x and x square e to the power mu x sine omega x and so on and so on. So, with multiplicity r you will go up to x square r minus 1 power. So, this way if you determine the solutions n solutions of this polynomial equation then from that you can determine n linearly independent solutions of this homogeneous differential equation. Now, after having the solutions in this manner you want to find out the solution corresponding to the solution for the non homogeneous equation with this whole thing equal to some r x. Now, if the equation is with constant coefficients like this and r x is one of those chosen function for the right hand side that is one of those special functions polynomial exponential sinusoid or their mutual combinations then the method of undetermined coefficients will give you a solution that is for this. So, method of undetermined coefficients will work exactly as the extension of the second order case. Here also those things those modification rules will apply that is even if this happens to be e to the power gamma x, but then if e to the power gamma x happens to be already there in the basis for solution of this then rather than proposing e to the power gamma x you will propose x into e to the power gamma x. Similarly, if e to the power gamma x x e to the power gamma x and so on up to x to the power r minus 1 e to the power gamma x up to this if all of these are setting already in the basis for the solution of this then even if e to the power gamma x appears in the right hand side function you will have to propose as y p x to the power r e to the power gamma x that is the next into some constant and so on. So, this is a general rule now. So, method of undetermined coefficients is something which will work exactly as in the case of second order equation and as you know that in the general case with general right hand side function and functions appearing in the place of coefficients a 1, a 2, a 3 this will not be sufficient. For those purposes for those situations you will need the more general method which is the method of variation of parameter which will work in the general case that is in this case that will work in this case as well. So, for that for solving that we will consider a linear combination like this. Now, let us so we propose a function of this kind which is a linear combination of all these n functions y 1, 2, y n which are known to be the basis members of the corresponding solution of the homogeneous equation. The basis members of the solution of the corresponding homogeneous equation that is the solutions of this. So, n linearly independent solutions of this equal to 0 is y 1, y 2 etcetera. So, y p is proposed like this. Now, we differentiate this and eventually we will differentiate it n times and insert all the functions and derivatives into this equation in order to find the solution find a particular solution of the non homogeneous equation. Now, as we differentiate first time we find that the derivative of these n terms will get us two n terms. In the first n terms we will have u i prime y i and the other n terms will be u i y i prime. Now, we try to put additional condition that this is 0. So, when this part is 0 then the first derivative of this will turn out to be as u i y i prime. Now, this is 0 means basically that the rate of the function u is orthogonal to the vector function in the vector space sense that is if u of x is a vector function with components u 1, u 2 etcetera. Then, u prime will be its derivative its rate and that rate is orthogonal to the vector function y of x. So, that is the condition we have imposed and that gives us y p prime which is like this. Now, note that this condition turns out to be the first row from this matrix vector equation y 1 into u 1 prime plus y 2 into u 2 prime and so on y n into u 1 u n prime equal to 0 that is essentially this condition. Now, imposing this condition we get the first derivative as this which is free from u prime terms which means that its derivative will be free from u double prime terms. So, we find out its derivative and as we try to put its derivative in that there will be again 2 n terms n terms together are taken as 0 that is the second condition that we have imposed. So, as we impose that from the derivative of this u prime y prime terms the sum of those terms vanish that is the condition we impose and the rest of the 2 terms u into y double prime that remain here. Now, this condition is equivalent to the second row that is sigma y i prime u i prime y 1 prime into u 1 prime plus u i 2 prime into u 2 prime and so on is equal to 0 that is the second row in this matrix vector equation. Similarly, we go on differentiating the derivatives further at every step we go on imposing these conditions till the n minus 1 s derivative that gets involved here and corresponding n minus 1 s derivative we get here and this is the point this is the stage where we have to find the last derivative that is needed that is y n y p n. So, when we differentiate this we get the full derivative in which full 2 n terms come u i prime y i n minus 1 plus u i into y n. So, these 2 both the terms will appear here in the n s derivative. So, note that from this point till this point there are n minus 1 conditions that have been imposed with the function the first derivative second derivative up to n minus 2 s derivative. So, these are n minus 1 conditions that have been imposed these are the top n minus 1 rows from here involving in the coefficient here up to the n minus 2 s derivative. The last equation we will get when we insert this y p n this expression and all these expressions for the previous derivatives and the function itself also. So, when we put all of these expressions now into this differential equation given then what we do we add to this last equation p 1 times this plus p 2 times the previous one plus p 3 times the still previous one and so on. Finally, p n minus 2 times this plus p n minus 1 times this plus p n times this that is from here. As we do that and collect the terms together then this term is here this term is here u prime y n minus 1. And here the other terms that is this plus p 1 into this plus p 2 into previous one and so on. So, there you will find that u is common in all these even up to this. So, in the that is why u i has been kept common outside and inside the bracket you have y i plus p 1 into the y i n minus 1 s derivative and plus p 2 into y i n minus 2 s derivative and so on that is this. Now, we know that each of the functions y 1 y 2 y 3 y 4 etcetera are solutions of the homogeneous equation corresponding homogeneous equation that is this equal to 0. And therefore, for every i separately this bracketed term vanishes. So, this whole thing goes to 0 this remains on the left hand side and r x remains on the right hand side. And that is why we get this last row from this system of equation. And what is the matrix sitting here? This is our old friend that is the fundamental matrix capital Y that we constructed from the state vectors of these n linearly independent solutions of the corresponding homogeneous equation these are the columns. So, here is our fundamental matrix capital Y of x. So, this is the last equation and this is the assembled matrix vector equation together Y of x this matrix into the vector u prime is equal to this vector with all 0s and r x at the last entry that can be said as e n into r x. So, it will mean that last one is 1 and here we have r x it is the same thing. So, this is the e n that is the last natural identity member in the n dimensional space. So, now this is the matrix vector equation in the rate u prime and this matrix is non-singular because these solutions form a basis that means they are all linearly independent. So, this matrix is non-singular as it is non-zero. So, this non-singular matrix can be inverted and this matrix vector equation can be solved. Now, usually I have told you earlier that in the context of linear algebra that the computation of inverse numerical computation of inverse with the help of this formula is typically inefficient, but here we are doing the analysis in terms of not numbers, but in terms of expressions and in this context this is formula turns out to be of advantage. So, inverse of y is adjoint by determinant. So, if we use that then the adjoint by determinant will come here and we can calculate u prime x. So, that is 1 by determinant into adjoint into this side. Now, note that this adjoint matrix will be multiplied with a vector immediately which has all its entries 0 except the last one. So, from the adjoint only the last column is needed that is adjoint y into E n will essentially mean the last column of adjoint y. So, that means when we construct adjoint y we do not construct the entire adjoint of this matrix, but we need to construct only the last column of it. So, last column of its adjoint will mean will require the co-factors of only the last row, because adjoint will be the matrix formed by the co-factors of this matrix transport. So, the last column of the adjoint will be formed by the co-factors of the last row. So, last column of the adjoint matrix is what we will need that is adjoint into E n. So, for that we will find out the co-factors of the elements from the last row only. Now, what is the co-factor of this entry? For the co-factor of this entry we will need to remove this column and this row and whatever remains. And for that we have to find the determinant and then we will put the sign according to the row number column number of this that is as which is the row number n and 1, 2, 3 that is the column number. So, that means the position of this will reflect in the sign and other than that this column and this row will be removed. So, the co-factor of this element we can get by replacing this column with 0, 0, 0, 0, 0, 0, 1 because this column anyway will be removed. So, if we put 1 here then as we remove this row and this column the rest of the determinant will give us the correct co-factors with the sign that we will get from the position. So, that means if we put if we replace this third column with E n that is 0, 0, 0, 0, 1 here then the determinant of this itself will be that co-factor because the in the determinant computation through this column we will get only one term which is 1 into the rest of it which is the correct minor and the position we will get from the expression of the determinant anyway. So, that means that the entries the first second third entry will be given as w 1, w 2, w 3 when w i is the same Ronskian same determinant of this matrix when evaluated with E n in place of the is column. So, this gives us the expression for u i prime and then when we find that this is a function of x only then we can evaluate u i as direct quadrature, direct integration and then this set of u 1, u 2, u 3, u 4 when inserted in the original proposal gives us a particular solution of this non-homogeneous equation. So, this is the way we develop the general solution for a non-homogeneous equation that is we first develop the general solution of the corresponding homogenous equation by replacing the R with 0 from that we find out the basis for all solutions of that homogenous equation that basis y n, y 1 to y n we use to propose a solution of the non-homogeneous equation including the R x and then based on this and based on that means based on this expression which determine the proposed coefficients, coefficient functions u 1, u 2 etcetera and then this gives us one solution of the non-homogeneous equation and then y 8 plus y p gives us the complete solution for the non-homogeneous equation. Now, many of these issues that is the Ronskian for the higher order or the basis for describing the solutions of the homogenous equation and then the way these basis members can be used to find a particular solution of the general non-homogeneous equation all these will find similar analogous situations when we discuss the solution of systems of ordinary differential equations in the next lecture, but currently in the remaining time of this lecture we will take a small digression into another topic which is in a way complementary to this classical framework of solving linear ordinary differential equations and that is the paradigm of Laplace transform method. Till now in the classical framework of solution of ordinary differential equations we have typically discussed with the understanding that the entire differential equation is known in advance that is both the left hand side which has terms related to y and its derivative and the right hand side R x. So, the entire differential equation is known in advance this is the typical understanding with which till now we have developed the methods and the typical way to look for the solution in the previous few lectures including the current one has been to go for a complete solution first and after getting the complete solution then if the conditions are available initial or boundary conditions then we can impose those conditions to find out the arbitrary constants arising out of the solution process. Now, this is the typical classical perspective however there can be practical situations where you already have a plant you already have a physical system which means that you know the left hand side of this differential equation completely. Now, that particular system that particular plant can be operated with different kinds of input which means this R x can vary from one application to another that can be a practical situation. So, you have the plant or system means that its intrinsic dynamic model you know and further matter you know the starting condition that is from which initial condition what is initial position what is initial first rate second rate up to n minus 1 s rate. So, all those things are known now as you decide to plug in different kinds of inputs you will be basically changing the right hand side. So, that means that as you drive the plant with different kinds of inputs on different occasions you will be changing the right hand side without changing the left hand side of the differential equation and without changing the initial condition. So, that means that entire differential equation is not known in advance only the left side of it is known the input function side is not known in advance that can change from one application to another. On the other hand the point of using the initial conditions afterwards does not sound so nice because we know the initial conditions beforehand. So, in such a situation another way of solution which takes care of the left side of the differential equation and the initial conditions first in one framework and depending upon the right hand side the last amount of last part of the work can be accomplished differently in different situations that is a paradigm shift. So, with the left hand side of the already known and initial conditions are known up to the right hand side r x changes from task to task. So, Laplace transform method give you a method which takes care of this part completely and keeps the job half done waiting for the right hand side to appear at any time. So, as a different right hand side appears every time the solution can be changed by another little amount of additional work. Apart from that there is another question that may be asked which is answered properly and adequately by Laplace transform method and not so well by the method that we have been discussing till now. And that is suppose this r x is not continuous till now we have been relying upon the existence and uniqueness result which use the notion that all these coefficient functions and r x are continuous and bounded. Now, suppose the input function with which we try to drive the plant try to drive the system is not continuous. So, for example, when power is suddenly switched off or switched on the if the power if the plug is giving you the input function r x and as it is suddenly switched on or suddenly switched off there is a discontinuity at r x say x is representing time here. So, as power is switched on or off what happens what happens is the question that we ask when we pose an initial value problem. So, this initial value problem is it well posed. So, we are asking this question that what happens what is the future evolution of y of t y of x in this context. Now, when we ask is this question is this initial value problem well posed then we say that does a solution to the differential equation exist or does the solution is the solution unique and so on. Apart from switching on or off if there is a certain voltage fluctuation suppose the voltage is somehow related to this input function input function of time. So, with time as the voltage fluctuates in the appliance in some appliance the model the dynamic model of which is sitting here then we say that the equipment the appliance which is connected to the power line what happens to its state how does the state evolve with time. Now, asking does anything happen in the immediate future that is the mathematical the mathematical equivalent of this question does anything happen in the immediate future is basically the question that for this initial value problem does there exist a solution. Now, this question is name because something will certainly happen. So, as something certainly happens we say that this initial value problem certainly has a solution. Now, depending on the R x being continuous and bounded may not help to find the solution in such situation. So, Laplace transforms provide a tool to find the solution in spite of the discontinuity of the right hand side function R x through certain ways of handling discontinuity. So, let us quickly have an overview of the Laplace transform technique the main salient features of it before we proceed to the system of ordinary differential equation. Laplace transform happens to be one particular kind of integral transform of this kind where there is a there is an integral there is a kernel function and f t function for which we are looking for the transform. So, this resulting function t of f t is actually not a function of t because integration with respect to t has been carried over carried out to get that transform function. The transform function will have the other variable s which is typically called the frequency variable. So, with the kernel function like this and limits of integration from 0 to infinity we have what is called the Laplace transform defined by this formula in which case this integral this improper integral is evaluated in this sense. Now, under certain conditions a function f t has a Laplace transform and the corresponding inverse function is called the inverse Laplace transform of a function of the frequency variable f s. So, if f s is the Laplace transform of f t then f t is called the inverse Laplace transform of f s. Now, with some background work people have developed some long tables of Laplace transforms and inverse Laplace transforms which we can keep as reference. So, right now I am omitting the details of this and going directly to the typical methodology of five solutions solving differential equations with the help of the Laplace transform method. So, suppose this is the linear differential equation with constant coefficient and these are the initial conditions second order differential equation. So, two initial conditions will be needed. Now, if we take the Laplace transform of both sides of this using the rules for Laplace transform of derivatives and in that how initial conditions appear etcetera. We will get the Laplace transform of both sides of this equation in this manner s square plus a s plus b y s where y s is the Laplace transform of the unknown function y this is what we want to determine and on this side we will have a plus a k 0 plus k 1 plus r s. Now, up to this point we can do except for this r s even without knowledge of this right hand function r t. The important point to note here is that this differential equation has been broken down to an algebraic equation and from here we can solve for y s in terms of the other quantities. So, this whole thing divided by y s will be the solution that is rather than the solution it will be the Laplace transform of the solution. So, even without knowing the solution y t we know the Laplace transform of it y s. So, then the next step will be after knowing r s also if we know the r s also then this whole thing together that is the inverse of it this is q s. So, that is why q s has been put here. So, then as we solve for y s from this algebraic equation of in capital Y s then we have the Laplace transform of y t. So, it is inverse Laplace transform turns out to be the solution. So, in this the initial conditions have been involved from the very beginning and inversion of the Laplace transform is conducted at the end to resurrect the solution. Now, in this framework of course, we can handle only limited number of plants with constant coefficients and so on easily, but then we can handle discontinuity of r t through two important functions and their Laplace transform. So, this is the unit step function if t is greater than a less than a the value is 0 if t is greater than a then its value is 1 and its Laplace transform has been determined and it is found to be this. Now, if a function appears with a time delay after time a then this is its effect and for that this can be the detail expression and for which we can work out the Laplace transform in terms of the Laplace transform of the original function that is what is the shift in time that is the result of the shift in time. This is one important discontinuous function for which we have the Laplace transform to be used whenever this kind of a this kind of an input appears this is suddenly applied and then after application that value is known. Another important discontinuous function is this derive delta function suddenly there is a huge jump in the value of r. So, and that value is suddenly at that particular moment the value is infinite very large every at the all other time it is 0 and with the special property that under that curve we have under that huge splash we have the area as unity. So, this is the derive delta function or unit impulse function. So, for this also the Laplace transform has been determined and that turns out to be this. Now, with the help of these two discontinuous functions and their Laplace transform we can handle the situations where the right hand side function of the differential equation is discontinuous. So, through step functions and impulse step function and impulse function Laplace transform method can handle initial value problems with discontinuous inputs as well. Now, another important term is there which appears quite often in the discussion of Laplace transform method and that is convolution. Convolution of two functions is actually a kind of a generalized product of two functions which is defined in this manner. Now, you can show that the Laplace transform of the convolution Laplace transform of the convolution is actually the product of the corresponding Laplace transform that is this. So, if you have two functions f and g and their convolution of is 8 then the Laplace transform of the convolution turns out to be the product of the two individual Laplace transform. This is called the Laplace convolution theorem. Laplace transform of the convolution integral of two functions is given by the product of the Laplace transform of the two functions. Now, quite often what happens is that in this context we know q s r s and we know that the moment we say in this particular case if the initial conditions were 0 k 0 and k 1 then these two terms will be 0 and this whole thing will go off and that means after this step with k 0 and k 1 as 0 we can invert this and find out q s and that is that gives us y s as q s r s. Now, rather than finding r s that is after the input function r t has been specified then rather than finding its Laplace transform r s multiplying that with q s and then taking the inverse Laplace transform one could do well to directly find out the convolution integral of q t original q t and r t that is because the inverse Laplace transform of q s r s which we want which we want will be the same as the convolution integral of the original functions in time q t r t. So, in that sense in many cases when the r t changes then rather than waiting for r t to appear and then compute r s quite often from q s itself the transfer function itself this is called the transfer function. From the transfer function itself we find out the corresponding q t through its inverse Laplace transform and keep it and then the moment a new input function r t is supplied then we do not go into Laplace transform further. So, rather than going to Laplace transform to find r s then multiplying this and then taking the inverse Laplace transform we can simply take the new r t and the originally earlier determine q t and construct their convolution through the definition of the convolution integral. So, this is another important issue which appears in the analysis of Laplace transform method and this will again appear when we later study Fourier transform. So, with this little discussion on Laplace transform we will continue in the next lecture to the solution of ordinary differential equation systems. So, ODE systems we will take up in our next lecture and from there we will discuss the stability of dynamic systems which will be solving through these methods. Thank you.


I literally am gonna tell her that I have a kid like I got another go pregnant I have something to talk to you about we're gonna get through everything unless you cheated on me. I Got a phone call today. Okay. I have a kid It's not your child. You got somebody else pregnant. Don't be mad. No don't touch me. Who have you become? You jeopardize everything for a fun night Oh What is good mom's welcome back to our channel if you are new make sure to go down hit that subscribe button turn on post notifications because we got a freaking banger today guys we're gonna be pranking Mariah Yes, I'm back at it again as you saw in our last video I told you I'm gonna be coming back with more pranks as you can see by the title I literally am gonna tell her that I have a kid like I got another go pregnant and I have a baby on the way And we're gonna see how she reacts to this. We're gonna see her blow up go crazy I think this is just gonna be off the charts one of my best pranks I can already see it coming so get ready for this video guys I want to make this intro really quick because we got a package in the mail and to get a package at our apartment You have to go downstairs to the other building and get it So it takes about like five ten minutes and I brought the camera along with me So I could just do the intro in the stairwell right now right before I go up to our room Which is one level above us right now so guys I cannot wait to do this I've literally been thinking about this ever since I saw Sam and Jess did this on their channel shout out to them I think this is a great video and this is gonna get it really good. So get this video thumbs up We're trying to get this video. Let's let's get it to 17,000 likes like this video for getting her again because sorry I'm stepping on leaves. There's leaves literally everywhere. So sorry if it's loud, but one more thing I'm not actually gonna be setting the camera like it's a prank I'm gonna actually be setting it up and like doing like a sit-down video.  And I'm just I'm just gonna be going for it guys. I cannot wait. I hope you enjoy once again I'm just so excited. I had to go up now. So let's go. Let's see what she does and hopefully I have a place to Sleep tonight. Hopefully she doesn't kick me out of the apartment. I'm getting so nervous guys now's the time It's happening right now. I'm walking in. Hey, babe. I Got the the package. I need you to come over here really quick. I have something to talk to you about and I want to film it Yeah, well, I'm actually I'm about to start filming right now. You want to come over here? I'm so scared I'm just so scared. I don't know how you're gonna react to this Let me start the camera. I had to film We need to talk No, it's not So I've been like depressed all day today because I got a text message from someone this morning and I need to I need to Tell you about something you're scaring me. What happened? I'm not back it up with you or anything I'm just I hope you don't break up with me. Why do you have to film it? Cuz I just want to share what's been going on. Is that not a good thing to do? Babe, you need to like just Be there for me. I believe this is like what's wrong? What happened? Okay? Talk to me. I don't want to say promise me You'll be with me, but like we're gonna get through everything, right? I mean let you cheat it on me Did you cheat on me? Why are you laughing? I'm not okay. There is this is not a prank. I need to tell you about it's about child Huh? You're scaring me talk to me, please. I Made I understand. I just this is not a comfortable subject to be talking about but I have to talk about it because it's like For you, you know, I'm okay. I ever it is in what way, okay, so I went out like a week ago, right?  You feel my hands. Yeah, I don't sweat. Don't cry babe. Okay. I know where this is going I would never do this ever again. I got okay. I okay. What happened that night? I got a phone call today, and apparently I have I'm only showing someone in your action. Okay, okay. I have a kid. I have a little child on the way Not I'm not pregnant babe. I know you're not pregnant. It's not your child. No What what don't be mad? No, don't touch me. Mariah. What did you say? Ficking thing not through this. Well, not the fact that you just cheated on me. I didn't know what that's our biggest thing You know, I would never do that like, okay No, no, no listen you need to explain to me what happened. When did this happen like a week ago? I don't know. I wouldn't even like you got somebody else pregnant. Yeah, I She's taxing me this morning and said Have anything to say is the girl Do I have anything to say? I don't know what to say. I'm heartbroken. I'm heartbroken that you first of all cheated on me It is a big deal. This is a huge deal You realize what you did I guess it you just cheated on me. No, no, but I could ask if I can just have a baby No That is not fair to me at all. That's not that is not fair to me And that is not fair to that girl Bell. What that is not fair. Okay? Well, she doesn't come Seriously become no. I'm just saying she doesn't want it. So I'm literally okay. Do you realize how I feel right now? I know and that's why it's so difficult, but I okay, wouldn't you rather not like have me not lie to you I could just lie to you That doesn't matter Do you realize why would I ever be with another girl stop crying? It didn't happen You really think I got another right? Don't cry. Don't cry. So what is this? Maybe I'm sorry. I feel bad now Why would you prank me like that? I didn't you really scared me. Maybe I didn't cry like that right? I'm sorry. He really scared me. That wasn't funny. I know I saw that. I did not mean to do that I wanted to throw up. You know, I would never cheat on you. You're literally like I'm gonna marry you one day I'm sorry. I did not mean to make you sad like that But I was gonna lose you again, you'll never gonna lose me.  Sorry, are you okay? Oh my gosh guys, come on. That was a good prank. I am so sorry babe. I literally you're gonna make me cry I literally did not expect you to go to me so good guys. I can't even you got me so Mariah Okay, come here. Hug me.  You know that I would never break your heart like that I would never your mind forever Okay, I did expect her to get like this honestly. I did I literally walk myself into this I know and that's why I stopped the prank literally I am not that much in an able to keep the prank going So I think I I think I did a perfect job literally only for like two minutes of saying that Right when she was starting to get really really like oh my god This is real I had to cut it because I did not want to make you actually start bawling out then everyone would have hated me Literally guys, I love this girl to death no matter what like happens. We're always gonna be together But we're never cheating or never breaking out. Oh, dude. Literally. We almost are a bit We almost been dating for four years guys or a year's mark 20 seconds We've only been on you for two years So you guys only got to see two years of our relationship out of the almost what three and a half now Yeah, so it's a little over three and a half years and we've only seen two and half years of our only gonna get better I cannot wait to get like married. I cannot wait to get now. She's all happy guys I can't wait to have our own baby our baby our own baby. Yes, little marble baby It's okay. I forgive you. I'm sorry. Literally. Oh, so what I what I did when I went down to get the package right here I brought the camera with me I didn't intro in the stairwell and I said that I feel like this is gonna be one of my like best pranks to get you With I did definitely top the last one that I did if you did not see the last prank after the end of this video Make sure to go watch that but guys. I will say what team bill did come back This is probably one of your best pranks. You're always saying how team does never come back is always team Mariah, so I had to hit you with a bang today. This was definitely a banger for you She literally was the oh my gosh guys. I still feel so bad Let's talk about our outfits and how we're matching in our shoes. Oh, I want to talk about outfits All right, we'll show my outfit of the day, you know, okay skirt in my shoes Okay, I see you let's show bill. I got some roses on my feet copied mean how to obviously wear the I dig this the new look and look at these fire kicks guys. I am obsessed with you We're looking hot and guys I want to show you guys our Christmas tree our stockings. We also have lights We got all around Everywhere there's lights everywhere Yeah, but we have like cute little Candles centerpieces Okay, so if you have a cat or a dog or anyone that likes playing with the freaking Christmas tree We have been doing everything in our power to keep our cat snap if you're new to our channel We have a cat named staff out of the tree. He's a bad. We're even gonna plastic like saran wrap the tree like on the bottom But this is what we did so far guys. We literally aluminum foil under here trying to block them We put all the branches so but he still gets up in here. He climbs all the way up in the middle It's so annoying What we're gonna do is we're gonna actually put this clear wrap all around like the whole bottom So he can't even get up because I feel like a fish in a fish. He's gonna keep running into it This cat will climb up all the way to the top all the way to here In the middle of the tree you can't even grab him because he's in the middle of the tree the tree came with a remote Look at this. So you could do like a flickering, which I do number four. It's either number four or one four We turn off all the lights and we have all these lights on and oh the vibes are so good at night Part of it's all coming together guys. We have literally almost everything that we need to live now I think we're all set. We don't even have to get anything except food anymore. I love this place so much I'm gonna miss it when we move into our house even though I'm really excited because we're gonna move to Florida soon And I'm excited. Oh, but yeah, if you are also new guys, you have a lot of catch-up on make sure to watch all of our videos We're so happy to have every single one of our Marvel Squad here watching every video and liking it up because we're about to hit 1 million We're officially 70,000 away. We're officially 70,000 away. It is crazy It's coming up so quick and I don't know come the new year. We might be a million subscribers 2021 sounds so good because 2020 has just been a roller coaster of shenanigans And I really hope 2021 just turns it all around just turns it all around the only thing that's been great about 2020 is all of you and the growth all the videos we've been making all the comments that we've been seeing from you guys It means a lot every single one of you guys comments It goes a long way so always remember that even if we don't like respond every single one of you guys We read almost every single See everything and we love you guys and we are so grateful because without you guys We wouldn't be here and we wouldn't have this apartment and all this night You know like I just I just love this life that you guys have given us and I'm very blessed I'm always bring them along with us. Yes, and I will always appreciate it That's gonna go on to the question of the day I want you guys to comment down below when you guys subscribe if you guys remember the subscriber number what yeah Yes, actually know when you subscribe Or if you subscribe literally right now comment down like two seconds comment down five seconds Comment down when you subscribe five months ago a year ago And we just want to see like where you don't even think about you have to be subscribed us for a year Like if you have that's freaking awesome But if you subscribe to us last week like just put it down or today We just want to see and we're gonna give our end of the video post application shout out Yeah, I just love sorry. I started to go I just love how like the video like I had your emotions like down and now you're just like perfectly fine I that's what I love about frame videos like she really is just so cute now Anyways, what is the post on the case shadow goes to Gracie Tynan? Thank you so much for being a part of our channel. We see you and we love you We love you we love every single one of you guys if you want to post a big a shout out Make sure to be subscribed with post notifications on like the video and comment down below And that's all you got to do and we will read all your comments and pick a post on vacation shout out Alright, well, we love you guys. We're gonna edit here, and we'll see you in the next video But until then


If we're mass producing fruit punch, unless it's like some customized production of fruit punch, I would think would be a process cost, because all the end result would probably be all the same. We'd be using to maximize production, so probably not that. See yogurt. Again, I would think the same thing. I would think that we're trying to make all the yogurt the same, and once we get the process down, the system down, we can mass produce it and make it all the same and use a process cost. See seeds for planting. And a little bit, we might have to do something on it, but I would think that all the seeds are pretty much the same. We're going to try to make a system to make all the seeds the same and gather the seeds and have the same amount of seeds in a certain basket or whatever. And so I would think that would still be a process cost and not a job cost. And then Ease has small gardening tools. And as long as if they're small and they're all the same and we're making them or producing them, again, I would think that we would make those all in the process cost. So this is one. So I don't think it's that this is one that didn't use customization. If we saw the word customized something and we make customized something, then that's usually for sure going to be a job cost. This one is a general contractor, which by the nature of being a contractor is typically going to be different. So contracting is a little bit different the way the jobs are tracked, but it's a job cost system and all the jobs are typically different. So I think A would be the final answer. Let's go through it one last time. Which industry would use a job costing system? A, a general contractor. Next question. Cost accounting systems are A, job order costing and perpetual costing, B, job order costing and job hunting system, C, job order costing and specific identification, D, job order costing and process costing, E, job order costing and first in, first out. So let's go through this again using the process of elimination. Cost accounting systems are now all of them have job order costing. So I mean this first half is pretty much a given, so we don't really even need to read that anymore and we're really just saying, okay, what's the second, what's the second half here? And A says perpetual accounting. We may have heard perpetual accounting, I mean that's a thing kind of we've probably heard perpetual system before, so I'll keep that for now. C says job hunting system and that sounds just kind of silly. Job hunting system sounds like it's something that was made up because they ran out of options or something. So I think that one probably not it. And C says specific identification, we may have heard specific identification but that actually has to do with tracking inventory in a merchandising company typically where we specifically identify the inventory as it goes. So it's actually that's not going to typically be a cost, a typical cost accounting, it's more of a merchandising usually. D says process costing and again we may have, that sounds familiar so I'll keep that one for now and E says first in, first out and again we probably heard a first in, first out but it's not usually a cost accounting system, first in, first out is how we track again like specific identification in a merchandising company typically. It's some kind of cost flow assumption. So I don't think it's going to be first in, first out. So let's go through this again. Cost accounting systems are job costs and what's the other one perpetual costing. That's kind of alluding to perpetual versus periodic which has to do oftentimes again with a merchandising company has that option. So it's not going to be that specific identification has to do with typically how we're tracking the inventory in a merchandising company and then of course it's going to be the process costing. So it's a job cost system and a process cost system. Those are the two that we're always contrasting when we think of a job cost system or when we think of a job process cost, whichever one we're working on we're often going to be asked questions or thinking about its opposite or the other one comparing and contrasting to the other one. So we're always thinking between job cost, process cost, the difference of course being the differentiation typically in inventory and other differences that result from it.


everyone. It's your girl, Kenzie here from Leighton Sports Cards and I'm ripping 2021, 2021, Panini Illusions basketball team all addition 10 box break number nine tiered random teams. And here's a look at the random results. You can always find this at leightonsportscards.com underneath the results tab. Link's also in chat for you. Best of luck. Here we go onto some basketball tonight. Tray Jones, Josh Jackson, got a LeBron. Nice. It's the impact Jay Sean Tates, Chris Paul, Cole Anthony, and Ovi Toppin. Jada McDaniels rookie, Devon Vasell rookie, RJ Barrett. It's going to roll the dice here. One, two, three, top, four, five, six, bottom. That'll go for the rest of the break unless it's a one of one. That is a five. So the rookie reflections will go to the bottom team. Isaac O'Coro on that one. Kawhi Leonard, Shining Stars, Bryant's, Ray Trent Jr. and a LaMella ball rookie. The wardens, Peyton Pritchard rookie, Sarah's Red rookie, Siddique Bay on the exclusive team all. Nice one there. RJ Hampton, Savon Lee. Last pass, pass one. Denny Avdea rookie, Kenyan Martin Jr. rookie, Ennis cancer. Amazing acetate Kawhi Leonard. Okay, Tee and James Brown. Got Jalen Smith, clumbly. Rookie reflections going bottom. Vince Carter, Josh Green, Robert Woodard. Precious Achua rookie, O'Kong Wood rookie. Got living legend Steve Nash and Reggie Perry rookie. That's Isaac O'Coro rookie, Terrence Ross, Red, Seth Curry. Going to go bottom, Colin Sexton, Siddique Bay, Erin Naismith and last pass, box two. rookie, Devin Dotson, Cole Anthony, Fox, Highlights, Ant-Man. Clear shots acetate, Jason Tatum manual quickly. Obi-Top and rookie, Giannis, Jordan Alora, Malachi Flynn rookie. Nice LaMella ball on the exclusive. Nice one there. Hornets, RJ Hampton. It's going to go Lakers bottom. Shining Stars, acetate LeBron giant, Hale, Isaiah Roby, Sabin Lee rookie. Culver, we've got Red, Malachi Flynn for the Raptors and Zabuki and Paul Reed. Moses Brown rookie, Jay Sean Tatte rookie. Amazing acetate, Anthony Davis and Pokozowski. Again everyone, Force will be doing the random for the LSE Nationals promotion. Ended all the spots for tonight. Tonight was the last night to get entered into that and we are all full so that is all closed up and he will do that random for group breaks tomorrow starting at 9 30 a.m. Trey Jones rookie. So best of luck to everyone who has bought a spot in a national break. Celtics on that one. Got Cole Anthony and Obi Toppin.  Well, Gina McDaniels rookie. We've got Devon the Cell rookie. What up six? He's six. Shining Stars acetate Kawhi Leonard and LaMella ball rookie. Hornets doing well with LaMella cards. Pritchard, Red rookie, Najee Marshall Pelicans. We'll color match there. Yes, sir. Jonte Murray. RJ Hampton and Saban Lee. Last pack box for him. Denny Avdia rookie, Kenny Martin Jr. rookie. It's going to go nuggets on that one. Amazing acetate, Kawhi Leonard and Moses Brown. Personals are also full tonight. We're going to get a lead start on that since we have to rip a couple of the breaks from last night's added on to the end of the schedule due to, you know, us losing Wi-Fi basically all night yesterday. Having to move some breaks around Xavier Tillman rookie, Kelden Johnson. I got the red and yellow exclusive career lineage Russell Westbrook. That's a cool one. LeBron. We've got Raptors on that one. Isaiah Stewart, Moses Jones. Any bangers today? Yeah, Sam had a nice breaking debut earlier. Let me tell you. Composal rookie, Jaden Smith. Pull the bat now. Jason Tatum. Clear shots. Kevin Durantz on the exclusive. Josh Green and a one-of-one Davis Mills out of contenders football. Trevor Lawrence out of what was it, 50 or 25? Yeah, man, missed it. Precious Atua, Gordon Hayward, Red Rookie, Pokasapsky. Nice one there. Okay, C. It's kind of an emergency debut because of last night. Yeah, really? He wasn't even supposed to be here. Yeah. Be like that sometimes. Bokongbu rookie, Okoro rookie, Kyle Lowry. We've got Cavs on this one. Shining Stars, acetate, Steph Curry. And Cevique Bay. I wonder if that's how it's going to go down whenever I go on labor. There's no telling when that's going to happen. So, closer gets to that. You're going to see me being scheduled. And then, nope, Kenzie's not here. Having a baby. Let's hope it sounds that easy. We've got Tyrese Maxx, a nice one there. Sixers on the rookie. KPJ. Oh, hell yeah. Yes. J. Shawn Tate and Lamar Stevens. I give props to the women who do it all natural, but... My wife, my ex-wife did not, my daughter, Skylar Mace, Patrick Williams. So, she tried that the first time and second time.  And what a difference it was. James Wiseman Rookie. Yeah, I know my body and no, I can't. My pain tolerance is not very high. So, I'm not even going to try and do natural. I already know. Killian Hayes Rookie. Mark and Ann, we've got Red Bradley Beale, Wizards. I'm going to check on this one. Yeah, of course, I'll try. I'll try my best. We'll be pretty busy with the newborn, but for sure, I'll send pictures and stuff like that. Lamello Ball. Malachi Knights. Kira Lewis. Absolutely. We've got Anthony Edwards. There you go, T-Wolves. About time. Joe L with the T-Wolves. Snodgy Marshall Rookie. Maxie Hakim The Dream. Living Legends Astonished Tatum. And CJ LB. I'm pregnant, Will Bear. So, like, having a baby completely natural, like no epidural, no drugs, nothing. Some women like to do it that way. And I give props to them, but I could, I could never. Zeke Naji Rookie. Richardson Yolkich. It's going to go Bucks. Butchua and Okangu. We've got Desmond Bain Rookie. There you go, Grins. Isaiah Joe. Zion. Campazo. Clear Shot. Bradley Beale. Wizards. Zion and Isaac Ocaro Rookie. Tyrese Halliburne. There you go, Kings. Lonnie Walker IV. The Red Exclusive Miles Turner. Pacers. It's going to go Nets. Derek Rose. Devin Dotson. And Cole Anthony Rookie. Last pack, Bob. Seven. Yeah, book seven. Trey Jones Rookie. Jada McDaniels. Dee Rose. Shining Stars Acetate. Yolkich. And Obi-Toppin Rookie. Yeah, Dean, I wish it, I wish that was the case. Shoot. Tyrese Maxey Rookie. KPJ. Colin Sexton. It's going to go Nuggets Bottom. Shay J. Chantete. Lamar Stevens. Skyler Mays Rookie. Patrick Williams Rookie. Shayk Milton. McDaniels. Mystique Zaclavine Acetate. Alladipo. James Wiseman Rookie. There you go, Doug. Killian Hayes Rookie. Mark and Ann. We've got Red, Brandon Clark for the Gris. It's going to go Suns. Nathan Nights. Kira Lewis Jr. Anthony Edwards Rookie. Another T-Wolves hit there. Najee Marshall. Westbrook. Lamello. Highlights. Living Legends. Alajawan. And A.C.J. L.A.B. Rookie. Azebuki Rookie. KD. Isaiah Roby. Insta Impact. Patrick Williams. Nazee Marshall. Xavier Tillman. Paul Reed Rookie. Boku Rookie. Got LeBron Jain. It's going to go Heats on the Bottom. Amazing Acetate. Dearon Fox. Kings. Brunson and Compaso Rookie. Theo Maldon Rookie. Ingram. We've got Red Rookie. Jaylen Smith for the Suns. Highlights. Curry. RJ. We've got Jaylen Smith Rookie. And Prashisha Chua. Box Pack. Box 9. And we'll get into our final box. Haven't seen an auto yet. Zeke Najee Rookie. Desmond Bain Rookie. RJ Barrett. Mystique Acetate. Devin Booker. McCollum. Yolkich. And Okangwu. All right. Last box here. Jordan Noir Rookie. Rudy Gay. We've got Red and Yellow Exclusive. Season Highlights. Jason Tatum.  Almost look like a zebra. Like off color zebra. That one is to 25. Nice hit there. Celtics. Who's got the Celtics? Let's go round. Jason H. Tatum the 25. Very nice. Impact. Cole Anthony. Killian Hayes Rookie. And Anthony Edwards Rookie. Nice. Malachi Flynn Rookie. Asabuki Rookie. Siakam. That's going to go Grizzlies. Mystique Acetate. Collin Sexton for the Cavs. Hardaway. Najee Marshall Rookie. Paul Reed Rookie. DeRozan. We've got Red. Okangwu. Sweet Color Match there. Hawks. Curry Highlights. Holiday. Xavier Tillman. Kampazo. And last pack here. Again, no recap. No autos. Not much to recap in this. Boku Rookie. Theo Maldon Rookie. McCollum. Living Legends Acetate. KG. Brunson. And Jalen Smith Rookie. Alrighty everyone. That will do it for Team All Illusions. We've got more basketball coming right up.


Hi everyone how's it going? Today I wanted to talk to you about a hack let's say a hack dealing with your email and how to get less spam or possibly no spam. If you are working as a freelancer, chances are I mean if you're working anywhere chances are you're getting a lot of spam. I mean let's face it in this day and age you're going to get spam because there's so many things that you sign up for whether you're online people want your email address to sign up for this newsletter you need your email address to order this or to get that and when you go to a store they want your email address even you know I go to j crew and they want my email address for some reason and you know you see this all over the place and people are always asking for email addresses after a while you're going to start getting spam you start getting weird emails advertising random crap that have nothing to do with you usually they're trying to sell you something that you don't want you know it's stuff you didn't sign up for and so what happened in those cases what happened is one of the places that you gave your email address to either well if they're dishonest they sold your email address or you know it got published somewhere and or maybe you even published it something you know you put it up somewhere and so they have these I mean they have all this stuff nowadays but these bots that can go trawl and and figure out you know find all these email addresses and so they can send it to random people or I mean you never know you know someone security j crew j crew's security might not be very good and someone someone was able to hack in and get a bunch of email addresses nothing against j crew I mean I just came to mind you know and why do people do this because then they can get access to thousands and tens of thousands or whatever of email addresses and just send a whole bunch of stuff out to them and it's worth it to them because it costs nothing to send all these out in fact they usually have programs that do it all automatically and you'll just get a certain number of people that'll click on the link that'll do this do that and we'll get some people to pay them money for whatever it is they're spamming you about but for the vast vast vast majority of us it's just annoying we get these spam emails and we have to delete delete delete delete and we unsubscribe we do this do that but that doesn't help anything so you need to go to the source you need to nip it in the bud first of all is if you have your own domain because I know a lot of you out there have your own domain and so I want to address that one first and the second one is if you don't have your own domain and you're using say gmail and um and you want to limit the amount of spam if you have your own domain your name is joe schmo and you you have joe schmo.com right then you make your email address joe at joe schmo.com so what you can do when you set up your email address and you can do this with all hosting clients no matter what you use go daddy register you know this that and the other you um you can make your email address a catch all account and what does a catch all account do well it makes sure to catch all the emails to your domain so maybe your email address is joe at joe schmo.com maybe someone misspells your name instead of joe he writes joe at joe schmo.com in which case you wouldn't get the email but if you have a catch all account you'll still get it in fact if someone writes to info at joe schmo.com you'll still get it someone writes to contact at joe schmo.com you'll get it someone writes to blah blah blah blah at joe schmo.com you'll get it you get the idea A catch-all account means any email that's sent to anything at joshmo.com, you'll get it. Why does this help? Why do I mention this? Well, because when you do sign up for stuff, say people at J.Crew ask you for your email address, then what you can do, once again no offense to J.Crew, I just came to my mind, what you can do is just write J.Crew at joshmo.com or whatever you want, J.Crew downtown at joshmo.com or something like that. The reason you do this is A, if they send you an email you're interested in, then you will actually get it. But B, if you start getting spammed a lot, if they spam you or if suddenly after a couple months you get random spams from random spam emails from random people and you see that they're all being sent to J.Crew at joshmo.com, well then you can decide to block J.Crew at joshmo.com and you can filter out so any email to that specific email address gets blocked out. This is super useful, as you can imagine, when you're getting spammed. Oh, Cardelmar. I rented a car from a company called Cardelmar. They're some car rental brokerage company. They were terrible. But so what I did was I wrote Cardelmar at mydomain.com. If I did get spammed from them, then I would see that the spam is arriving at Cardelmar at Akahi.com, which is my domain. And so I could then block everything that comes directed to Cardelmar at Akahi.com. You can do the same thing. You have joshmo.com, J.Crew is sending you spam. You can block anything that comes to J.Crew at joshmo.com, but you'll still get all the stuff that's sent to joe at joshmo.com, at info at joshmo.com, at Macy's at joshmo.com, whatever else he signed up for. So if you have your own domain, you should get a catchall account and use a different email address before the at sign, because that way you can control the spam later on when you get it. Now, like I said, some of you don't have your domain, and you'll be using one of those free email services, usually Gmail. Pretty much everyone has Gmail. So what you can do with Gmail is, once again, if you're there at J.Crew and your name is joshmo at gmail.com, what you can do if they ask you for your email address is write joshmo plus sign J.Crew, everything attached, joshmo plus jcrew at gmail.com, and you'll still get the email. It'll come to your joshmo at gmail.com, but it'll also be personalized. So that way, if you start receiving spam to that specific email address, you can always just block that specific email address, and you won't get any more spam, at least not from that source. So this is just a way in which you can decrease spam. Unfortunately, because if you start getting spam to your main email address, you know, you have to delete them one by one, block them one by one. But if your email address has been sold out there somewhere, you're going to have tons and tons of people with your email address emailing you. But if they have an email address that you can block altogether, I mean, then it's great because then you can block that email address and go on with your day. So anyway, I hope you found this useful and I hope it made sense. If you have any questions, feel free to let me know. But this is something that I think all of you should be doing with your email.  Nothing's 100%. Some spam always comes through in this day and age. It's pretty much impossible, but it really does limit the amount of spam you get. And you can also filter out any spam you get right at the source. So you don't have to bother with it for years and years and years. So I hope you found this useful. If you did, please don't forget to click like because that always helps. And don't forget to subscribe for more emails that can help you out as a freelancer, as a freelance translator. And also if you do subscribe, if you want to be notified when you get a new, when a new video is out, then after you subscribe, you can click on the little bell next to the subscribe button. And that way you'll be notified and you don't have to check to see if and when a new video is out. But you'll know right away when there's a new video out and hope, and there'll be more tips and tricks for all of you freelancers out there. So that's pretty much it. And I'll talk to you next time. Thanks. Bye.


My name is Becca McCarron-Tran, and I am a fashion designer and founder of the Future Forward Bodywear Label, Chromat. Ever since the beginning, Chromat has celebrated all different types of bodies on our runway shows and campaigns, so we're one of the first fashion designers to show plus-size models at New York Fashion Week. We've always had trans models and queer people in our shows and campaigns and work with people behind the scenes equally. We represent different communities that have been historically underrepresented in fashion. We've done so many things over the years, but we've always approached everything with a sense of experimentation. I think coming from architecture, treating the body as a site to create augmentation and enhancement, I never look at the body as something that is wrong or that needs to be fixed. I guess that treatment of the body as a place to be celebrated, not a place to be fixed, is something that has always been a through line in what we do and I think separates us from other designers. I studied architecture and I worked for architects for many years, and Chromat kind of evolved as something I did on the side. It was a way for me to experiment with design and basically take the design process I learned in architecture school but apply it to a smaller scale, which is the body. The best thing about Chromat for me has been collaborating with other people and me bringing what some idea I had, but then it morphs and evolves and changes and gets so much better than I could have ever dreamed of. She really respects your integrity as an artist and as a collaborator and respects your vision, and so I think that's really important and that's why she's been able to establish this really beautiful community and family of people that really care about the people that they work with, care about their message, the meaning behind their work. Finding my authentic voice or figuring out who I was, it took a lot of dredging through that messy marketing media about how women or girls should present themselves and so I think starting Chromat, I knew from the beginning that I didn't want to only celebrate one type of beauty or how skinny white girls only in our shows. I understood the power of representation and the power of showcasing different beauty or showcasing people as they are authentically is a good thing and not everyone has to conform to a single way of being or presenting. So I think that from the beginning, I think I was aware of the dangers of fashion and the dangers of media representation, so I wanted to utilize Chromat as a platform and be able to rewrite some of that narrative.