Yo, what's going on guys? We're back with another episode of the after hours podcast today We have a special guest George who is a member in MIC. He's been with us for a little over a year now His name in chat is GSC Little over a year cool cool and he actually if you guys remember Joe Angelo We interviewed him a few weeks ago and this is his tab is trading accountability buddy So it's pretty cool to have him on too so we kind of hear both of their stories So thanks for coming on George. Yeah, man. Yeah shout to Joe Joe's a real one man. He's a cool dude. I like him a lot He's super friendly, so he's good to have an MIC but but enough about that bum So George, how did you kind of get into trading? How did you find MIC and and want to kind of lead us to that part of your journey? Yeah for sure. Yeah. Yeah So actually I found trading like in 2016 2017 with Tim Sykes and I'm not sure how I actually like found them or anything like that, but Yeah, I found him and It was really I was really just you know, I I never really You know, I never really traded You know when I was initially studying like in 2016 2017, but You know, I don't know I just I kind of just like followed like you know what he was doing and Yeah, I just study a lot like you know you know, this is a His courses and stuff like that, you know, I was in this chat and then I Basically stopped all that and 2020 basically when the pandemic started that's when I kind of got back into it It was with him again. So like I kind of just like you know joined his program and Yeah, you know just started learning trading like you know from from you know from his Online courses and stuff like that cool, but um, yeah I mean Yeah, it wasn't profitable at all, you know, like I I Never got lucky or anything like that Yeah, yeah, and as I mean, I was on Twitter as well and I forgot how I stumbled upon MIC. I'm pretty sure it was Twitter I'm not sure who it was. I think actually I think it was Harry. I found Harry Haas through Beyond the PDT With Bryce and Matthew. Yeah You know, I was following Jack and and Kyle Williams and all that stuff Yeah, those guys are all like, you know good friends of mine Like I mean what I think like not to like cut into your story or anything like that, but I think that when you kind of You know get to know people and like it wasn't necessarily through like I knew Jack Like before beyond the PDT and we were just kind of like going over because like I was a long trader He was like an OTC trader. I was like maybe I can teach you some stuff about longing You can teach me some stuff about OTC We can kind of you know mix and match about stuff like that But like we've really kind of like honed in on our own style But yeah, I met, you know, Monaco I met Bryce through Beyond the PDT and I got to know like a bunch of Sikes guys kind of as well and you know Some people who I talked to all the time some people who I don't talk to a lot But I think as you kind of you know get better in the industry you get to know some people like that and you know You know never anything bad to say about like those guys, you know, yeah great guys, so I know that was fun, I wish they kept doing it to be honest There's a lot of fun So George at this point when you found hair you found MIC Were you just like buying breakouts or were you just like did you not even have like any sort of process? Were you just kind of winging it and hoping it would work or yeah, what was that part of journey? I mean with the 2020 pandemic, you know, like all the coronavirus stocks, you know, it's so easy to get lucky like I guess there was a period of where I was consistently winning But there was really no like, you know process to it and yeah You know after like such a strong bull market, you know, you could really just buy anywhere at the beginning of the day and and and make money yep, and And it wasn't until somewhere in 2020 where the market started to slow down I basically lost, you know, I made a few thousand and then I just kind of lost it all back But yeah, there I just I had I had no process and like once I remember I would attend your guys's Wednesday webinars with Tosh Yeah, and, you know, I don't know, I guess I Had to be like, I don't know.  So what was your experience when you first joined did you find it was like Easy to navigate you find it was easy to like talk to people and like, you know, meet others and all that stuff.  So it was just yeah, it was just a big difference, you know, and I'm glad I took that chance and you know, I'm glad I'm here with you guys today and So much has changed, you know, like I just Yeah, just like so much Knowledge now of you know of what's going on and You know, but the MIC process and Yeah, so I don't mean to like cut you off or anything like that But So you join MIC and you know, you watch some of the webinars you kind of get to know you maybe watch because you're more long biased right Yeah, yeah, I really have this style like similar to yours. I would say Yeah, so maybe we can kind of dive in and like talk about that throughout the podcast as well And so but just since we're at kind of like the initial stages of just like talking about how you join and stuff like that How did you kind of go about like attacking things? Is there anything that you would do different like since when you first joined Is there is there, you know, like how would you kind of like attack the market now You know, those are kind of my questions that I was kind of wondering from you Well, I would say in the beginning I've always liked longing so and any teeth I've always loved your videos Harry and like I've always I would say like gravitate towards your strategies and stuff like that And yeah So I mean I would just you know Learn from the videos. I mean I would watch all of my CV is everything you know everything is helpful But you know, I would see your trades. I would you know You know see your commentary, you know, main chat is very very helpful and You know, I would use small size. I'm using bigger size now but definitely in the beginning you know it's best to use small size just to like test things and stuff like that And you know things to start to click you know and You start to understand you know You know why stocks move a certain way And you know you kind of start to see patterns and you know just like all technical analysis that you guys teach It just it all started to click you know and you just know you start to feel confident and Yeah, so like I'm very confident and you know for example like short traps. I know I know Harry that you love short traps you know the first bounds Just you know all the strategies that you know that you like I I definitely have learned and you know still still mastering but That's cool. So what what drew you to Harry style of trading because I know like obviously people when they first get involved, you know It's a tough decision between long and shorting and you know we have really good mods for both and what kind of drew you to to longing and what drew you to Harry style of longing particular I think it's because well I guess initially I have a small account so I I couldn't really short at least in the beginning So you know longing was kind of my only option at first But as I studied it more because I knew I had to at least study that first I actually started to like it and You know I've always studied shorting as well, you know when I When I study You know the MIC videos or Just you know with Tom, you know You know all short bias trades, you know you could learn From everyone even though you're alone You know so just having like multiple you know multiple perspectives of Both sides, you know that that could really help you so yeah for sure. That's great advice So since you're here and we only have like you know 30 minutes or so I try and you know jam it in and you know, you know, you know 40 minutes or so I try and you know jam it in as much as we can because There's so much to kind of like talk about so what are some things that you find that you're kind of struggling with I mean, you know, you have me here right now It's not like a pre-recorded video or anything like that or like a DM and not a robot What are some things that you're kind of struggling with now that maybe you need some clarity on or maybe you kind of like to You know kind of learn about or you know maybe talk about some struggles that you have with like maybe FOMO or like I don't want to put any words in Like in your mouth, you know because like I have no idea but maybe just talk about like some kind of struggles or like how you would kind of attack every day or just anything like that, you know Yeah, psychology wise or trading wise like anything anything involved with the markets Right, right. Yeah, I would say I would say it's like partly discipline. It's also patience.  You know, some days I do really great, you know, with my discipline and patience And just some days I just give into FOMO and And it kind of just ruins the whole day and I also have trouble With the concept of being under PDT Just the fact that you know you you basically have limited trades and Sometimes that puts pressure especially, you know, if you like like you hear you mainly trade like the first I would say well two hours, you know, yeah And And if you think about it, it kind of puts pressure, you know, like pre-market you're watching all the stocks, you know, watching it for hours and stuff like that and It comes down to that, you know, like those, you know, maybe one, maybe two key moments where you hit, you know, you have opportunities to take a trade Sometimes, you know, being under PDT, having those limited trades, you know, it could get to you So maybe I'll get in too early, you know, thinking I'm going to miss a trade or Sometimes I think that I'm patient I'm being patient and then once it gets kind of towards midday and there's really not that much opportunity left I end up like, you know, foaming into like a trade, you know, I shouldn't be in and stuff like that but Yeah, I would say patients, a little bit of patience, a little bit of discipline Eric, you want to talk this one first, then I can go into it I think for me, especially with longing, longing is something where not you're not going to see every stock go to the moon I mean with any stock, right, you're not going to see every single stock all day fade, you're not going to see every single stock go to the moon And it really comes down to putting rules in place so you're not putting yourself in a position where, you know, you're making those mistakes And so I think that I have different rules for everything.  So I guess you need to remember that every single person is really dealing with the same issues, but it's just really how you kind of go and attack things, you know, and, you know, being too early is is another curse for a long trader because When you're longing like you're always like, oh, I want to get in like kind of like, you know, when I was first longing, I would like long things that like didn't have any volume that were kind of dead, because I was like, man, if this thing runs this is going to like, you know, nine or 10 or whatever, but the reality is is that maybe only 20% or 10% of those types of trades are really going to work well for you, you know, And that's why in longing, you need to be just so so so kind of disciplined. And so my, my advice to you would really be, you know, if you see something pop up on your screen, and you know, you already feel that FOMO and you already are like, man, I got to get in this I got to trade this, you know, take as much as an hour, you know, set a timer on your phone for an hour and say I'm not going to long this for like an hour. And that can be difficult because you can see stocks that get overextended and that just keep going and going and going. And so after they top out, I usually wait, you know, like an hour. I think another thing is that, you know, as far as the revenge trading goes and especially with PDT, it's difficult when you kind of get into early you make a mistake and then you're like, I only have so many trades for the week, but you know I want to get this back. If some if you find yourself in that type of situation, it's best to just really wait another hour. You know, it takes sometimes 30 minutes to an hour for these emotions to kind of cool down and drain out of our system, and allow us to take another, you know, type of trade again.   You know I haven't got it now I'm going to wait like an hour 30 minutes I'm going to take some really kind of collect myself. And just one more thing is that I think stock selection is really key.  And also another thing you never want is that you never want to be longing things that really are not doing that much volume and really don't really have that much attention, you know, and you know there are times where, you know, like we talk sometimes and you're like oh well that stocks only up 20% or oh that stocks only up this much or it's that much. I never said hard and fast rules I'm only paying attention to the volume, you know how much attention I think this thing has. What's kind of going on because by the time something is up, you know 100%, sometimes it can't really go up any higher, and we hit this kind of like price ceiling, and we saw it on SNLA today where the stock just kind of tanked out of the open right. Now I'm not really someone who puts hard and fast rules on, you know, the percentage or anything like that I'm paying attention to is it the hot chick, does it have an attention are people looking to kind of get involved. And then like when we get that kind of dip. Okay, are we back up again, you know, do I think we have some short sellers who were kind of had that FOMO that you kind of talked about on the long side on the short side right. And that's how we kind of play against those types of people and get that move higher. So I think really setting rules for your FOMO if you just take a look at your trades and find out what mistakes you're making and you set rules to like my rules are designed so I'm not trading. I want to have so many rules that I only can make, you know maybe less than five trades a day right, because if you're making too many trades a day or you're not making the right trades a day. So I think that's something that can kind of hurt you on that road to consistency but that's just kind of my thoughts on that. Right. I think, I think for longs to this isn't even like part of my point but like I think for longs to like rules are so almost much harder and like much more important because I think it's a long it's like you, you it's a hard because obviously a stock and technically run from like $1 to $100 right.   There's discipline, like chasing, chasing like kind of following trading. Discipline patient and fellow. Yeah, so. So I guess on the first, the first part of discipline is and this is my favorite point to bring up to people is the thing with discipline is one we've talked about it before in the past where if you're not disciplined outside of trading it's almost impossible to be disciplined inside and I fully believe that. So if you're struggling with discipline, I challenge you outside of trading to put in things in place whether it's going to the gym daily diet, not drinking during the weekdays put something in place that you have to test your discipline outside of the market because you have to work on it outside of this as well, and that will translate and correlate kind of into your, your market behavior. I think the key thing with discipline is through discipline, you will find consistency right and what makes the best traders and people like Alex in or about or Tosh or any of these guys great is that they're just consistent with their discipline.  And through teaching yourself discipline outside of the market, bringing it in, you're just going to notice slowly over time like, wait a second, I'm following these rules that I have in place, and I'm not losing. And then the more you do that, in my opinion, the easier it gets to stay disciplined if that makes sense, because you just like it's the same thing in life in general like I work out or try to work out every weekday right. And the more I stick with that the easier it is for me to do that and then also do my diet and also maybe not drink on certain days or whatever because it's all around a focus in my life. And discipline stuff. It's not easy man discipline stuff because it's that's why trading is hard because I think 90% of it is discipline and like patients and all that. Now as far as patients. I remember Harry said something a long time ago and it was a really good kind of quote and he was just saying that the more patient you are with your entry the more patient you can be with your exit. And I find that really true for both longing and shorting because it's the same thing like, if I get in at that for especially shorting because I scale into my short sometimes like one or two bullets, longing sometimes I don't really, I'm something I think so I can't speak on that but for me it's like half the time if I just wait for my initial line and my initial entry, not only would I have a better average I can size in easier with less stress. So like what I do a lot of times is the very, it's a point from bow as well, is the first line that I want to get in at, I just skip, or I use like tiny tiny size like if I want to start my position with like 1000 shares and say, I might use 100 shares at that first kind of line, because I know that if I'm more patient for the best entry, then it's going to be make me more patient for my exit, and also give me more confidence as far as adding size and staying calm like in my, my position, because if you just size into the first line and you can't stay patient for that setup, then you're you're just going to end up either getting shaken out or it's not going to be the trade you want you only feel great about it. And honestly, if the more patient you stay for your setups, it kind of correlates back to your discipline, the more patient you are your discipline is going to kick in, you're saying all right, I'm waiting for my spot, and this trade is going to work.  Now, as far as PDT, I guess on this last piece to and like kind of like PDT revenge trading all that stuff.  Unless it's Canada like Eric. So, I think that message.   I had to avoid any setup that I thought was sub par. And at this point in your career, you need to identify which setups are sub par which setups are just like you're trading to trade because when you're over PDT you can kind of take those sub par setups with smaller size and and try it out and see how it works. But when you're under PDT you're at the table, you have one bet.  And I know it's hard and I know it's tough, but you have to why yourself to kind of understand that like you're this is the position you're in right now. And you will get out of it by staying patient right your whole issues are these issues is through concern are just related to discipline with everything right. Like if you stay more patient and more disciplined while you're under PDT you're going to get faster over PDT, because when you kind of break that and you revenge trade and you do something like that, you're pushing the you're just pushing the goal line faster and faster farther right you just need there's more money I have to make to get over PDT, and it's a huge bitch.     And you'll find that if you can avoid those little moments, the better setups will show up. Yeah, I would say sometimes it's easy to get tricked into thinking you know there's a confirmation of entry. You know, I just feel like you know when when when is the right entry when is the right confirmation signal and yeah sometimes you go in a little too early and what gets me is under PDT you know like, you know, say I have a few trades on the day, you know that that's one trade wasted if I get if I got into really even though I do have that gut feeling. Like you said, I don't know, you know, how you get that gut feeling but it's kind of like an experience telling you like hey, this doesn't feel right.  I can't do that yet since I'm under PDT. I know I should, you know, cut like for like break even.  Yeah, that's not easy to that's not easy.  And with longing it's so important that you're not chasing these massive pump candles or that you're not buying at the top way over VWAP or you're not buying into a whole number you're not you know, so all the all those things are very important. And so what you can do is that you know you say to yourself okay. I'm going to sit there and say to myself.    By the time you get to Wednesday I guarantee you're going to have a better trading day and say wow, it's not as tough anymore to follow these rules because I've started to make it a habit. By the time you're there in two weeks, and you have got you know these habits built. You know it's so easy, you know it really takes 30 days. You know to build a habit or 90 days or whatever, and it might take like, you know, a bunch of running around and foam on revenge trading after you've broken that rules to kind of start to break it and then you fall into a new kind of set of bad habits. So maintaining good habits is something that is definitely key and just actively being aware of like okay, you know these are, are what have, what have, you know, these are the goals that I'm striving to achieve this is what I want to become. This is what is going to kind of get me there I'm going to develop this into a habit, and that can help you a lot. And also, another one is just trading that like you don't need the money, you know, I see so many long time to buy the top of the range I see so many shorts trying to short the bottom of the range chasing. You know if you're chasing anything, and like I don't care what it is in life, you know you're not going to get it right, but if you're patient, you know you're sitting there you're waiting for it. That's something that is going to help you a lot I had a conversation with two other long guys today. And they're like man we sold too early, and then you know the next day they're stopped out because they didn't sell at the right time.  I'm selling always at the high day every single day. I'm just drawing the bottom I'm drawing the top I'm saying okay. Like, I find in this market inner lines have been working very very well for longing.   It's like, you just are at a point where you just need to just keep building singles and just one bill right one single snowballs into another snowballs into another snowballs into another. And it really gets you to becoming profitable. And the last thing that I just wanted to talk about was just take a little bit of time and wait for the cell, wait for the setup to kind of develop right if it's a VWAP reclaim you're looking for there's no need to buy that first view Yeah, and wait kind of like on a DB GI today. Yeah, cut you up if you if you kind of saw hovering around 450 and it looked like like you know it's gonna have like a little VWAP reclaim, you know, and I actually got into early and right when I got in I knew I was in too early but you know it just goes back to like the limited trades I'm just gonna set us up a little below. It's just an unnecessary paper cut. But yeah I recognize I was too early and yeah so I just wanted to And we can kind of talk about like that kind of scenario because that might be a good scenario to kind of end on. You have to be saying to yourself. Okay, in this chart.  So when I look at DB GI and I don't have the chart in front of me, but I believe we had 450 we had VWAP and the top was maybe like 5550 right. The top was already set on DB GI. The odds of that climbing all the way back and pulling like an EDSA from two weeks ago or something like that very, very, very slim. If I have to say something from two weeks ago or from last month, it's a setup that does not happen often. So on that I simply left that alone because I knew every single long trader did not get what they wanted today. ALF. Yeah, we got you know eight bucks to 840 you know I didn't feel that trade in my bones when I got it I wasn't like this is a great trade I was like this is a good scalp. You know, and I think for long traders they're always looking for home runs because you know you're longing and you have the home run potential right and that's what James kind of talked about like James knows like okay this stock can only really go down so far in this particular situation but with longing it can go higher and I can go to 10 and the moon and whatever right. Longing is very kind of optimist biased right. And so I think that in that kind of situation of DB GI right yeah we got over VWAP, but you have to remember VWAP is a resistance level of supply. What was right after VWAP 450. So not only do we have VWAP as that kind of supply area that makes every single long trader hesitant, we have 450 right after it. And 450 is a difficult difficult level to break, and especially coming from that you know kind of 430 420 area, you know from that's a long way for that stock to go. So in that type of situation I'd rather see it kind of get to 450, take a break, wait an hour, wait at 30 minutes, wait a while and then see if we can come back. Right, because I need more time basically 100% where are shorts going to stop out on that chart where do you think shorts are going to stop out on that chart. I would say, or pre market highs. Yeah, right. So, you know why do you want to be in a situation where you're longing, you know before no shorts and trouble right, and there's sometimes where we get to pre market high day too quick and we stuff and we go lower. That's a type of situation where the chart wasn't meant for that type of play right, but there are situations where the range is tiny and we can get that short stop out and we can keep moving higher but these are things that you need to really take note of of. Okay, shorts may be stopping out, but this chart has gone up a little bit too much for me and it's a little too over extended for me to want to get long on. You really have to be very, very, very selective in what you're longing and, and the way things go up in the way things move and, and how you figure that out is. I think you've got enough screen time by now it's really just saying to yourself, yeah, I am not going to be longing these overextended charts for these broken charts. And if you say that to yourself that'll help you a lot right dbgi. Like you can admit that was a broken chart right and you probably longed it and felt stupid afterward after a tank, but you need to be saying I feel like it wasn't really broken but I mean that. I mean, I feel like the chart was still in, still intact but. You know, when we're 50 cents away from hi day. You know, 60 cents away from hi day. What do you say, Harry, if the homies are shorten it at the top. I don't want to get caught in the drop. Yeah, we also had a left competing, right. So we had competing in that type of situation as well. So not only do we have a left, we have a heavy flow, we have a ton of supply ahead of us. Right. Just, yeah, would you say that pre market was kind of active this this pre market this morning. I mean we had SNO a wasn't a great one wasn't something wasn't a stock that I loved we had a left that was kind of moving around got broken got reclaimed. I don't know if it was I and yeah we also had dbgi I mean I guess it was active. For me, it was active but it didn't give a lot of great long opportunity. You know we had a lot of overextended things move higher, you know whatever and I'm okay. The thing is with me is that like I've made enough money now so if I miss something I'm like fuck.  And you've got to be like that go in order to really move on as a trader and really kind of evolved. That's one thing that you really need to let go as along it's tough because we don't get pre market runners every single day. Right, as a short we get a ton of AIDS pre market.   I mean, again, I don't think it's easier to do either thing. I just think there are a lot of kind of easier kind of lay up short sell up setups kind of in the pre market and stuff like that. It really depends on the market cycle right, but I mean, I don't know.  But when I see something go high from seven to $12 as a long, you know, that is something as a new trader that I felt a lot. And, you know, it's hard man. One thing I've actually been doing is I've been waking up. I mean, for a while I usually wake up over here and I'm in California so I wake up at like three around three o'clock in the morning. And, you know, I would get to the screens pretty early and I just I feel like I did that because I didn't want to miss anything. And lately, I've been starting a little, a little over an hour after that. That's what I've been doing. Yeah, so it's awesome. I think it does help because sometimes you know, if you, if you're, if you start with the loss in pre market or something like that. Yeah, so I've been waking up later, you know, to help with that.  Maybe that gave me a little, they gave me a little, you know, FOMO. Yeah, that's why I jumped into that. It happens man but it. You can do something I just want I don't mean to cut James off you can what you do is keep a little journal and what you can do is keep your emotional temperature in the morning and ask yourself questions. Did I miss something. Did I see something. Did I sell too early. Anything that could trigger you in the least to revenge trade you really need to seep into your emotions and get to know yourself so well, take your emotional temperature. I like that a lot. Right. Take your emotional temperature and gauge where you're at.  And I think that's that's a really, I like that a lot actually that was really that was smooth. That's cool and I think, I think we're coming up like on our time where we do have to wrap up but the last little piece of advice I want to give you before we do is, I really recommend for long buys guys to ensure it's anyone new, you know, all of trading in the beginning is just lack of a lot of it for people's lack of confidence in your setup. And something I did early on that was a really key piece was I really did print out and screenshot, all of my charts and Alex's charts, and even ballast charts and stuff and I actually put confirmed moves like next to my screen so like if I'm in a setup because I'd rather in the beginning, especially under PDT I waited for a move to confirm before getting in. So I just had that reference right next to me so as a long like you can like print a view of reclaim picture, put it right next to you and you know what it's going to what you want it to look like for it to be a confirmed move. Yeah. And, and that helped me out a ton that helped me out a ton just having that reference point like right next to me. Yeah, but yeah I know we are coming up on our time we let it go a little bit but it was actually really nice. Just curiosity. What are you doing right now like for a living, like for a job. Right now, I'm actually, I'm working part time with my dad he owns a business. And kind of just like side jobs with him and then I'm actually door dashing throughout the day here and there at night. Just so I can really, you know, have be able to train in the morning, you know, get that screen time and I've been doing, you know, for about a year now. And yeah, it's kind of, you know, it's kind of a weird schedule but I think it's definitely worth it, you know, and I definitely thank you guys, you know, for all of them I seek, you know, for teaching me and, you know, being where I am today so it's. Hell yeah bro. Hell yeah man well thank you for coming on thank you for talking with us and I'd love for you to update us further on in your, with your progress. Yeah for sure man. Oh man thank you so much George and thank you again. See you. All right thanks. Thanks guys.


Welcome back. We're at lecture 34. We're in the supplement to the textbook, which is second order differential equations. We're actually kind of scheduled to start a new section, but we've had a couple of web assign situations that might need to be addressed. One of them is one that I actually had as an example to show what could happen with the particular solution. And the other one, let's just take a look at that, at least enough of it where you feel like you can take it from that point. One of the web assign problems from 7.8 looks like this. Does anybody have that printout so you know which one it is? Number two. OK. They give you the complementary solution, or what we have called the homogeneous solution. We could figure that out. It'd probably take less than a minute to get the characteristic equation r squared plus 2r plus 1. If you factor it, that would be r plus 1, the quantity squared, double root of negative 1. So if we have a double root of negative 1, then we ought to have c1 e to the negative x plus c2 x e to the negative x. That look right? For that part, I think that's given to you, but certainly have the ability to come up with that very quickly. That's part of our answer. I think it's our job in this problem to find the particular solution. So we've got a generic linear term there. So if we wanted to kind of characterize what that is, that would be something of the form ax plus b would generate, have the potential to generate something like that. We're going to change this, but we could also kind of generalize e to the negative x. If we want to end up with an e to the negative x, we can start with a certain amount of e to the negative x's. I think we had a problem like this two days ago in class where the d could be distributed to these. We don't know what they are anyway, so it's really kind of redundant to have that. So we can kind of dispense with that. If you were to distribute this solution, or the proposed particular solution, you'd have ax e to the negative x plus b e to the negative x. Anybody see a problem? OK, aren't both of those up here? This is really this. We don't know what a is. We don't know what c is. So they're both the same. So that's really not going to be the particular solution. This solution is right here. So both of these pieces, it seems to not be enough, or we're not going to need an ax e to the negative x term. We might as well dispense with that completely. We're not going to need a constant times e to the negative x either. So we kind of throw that out at this point and try again. Well, if the x e to the negative x we don't need, and the constant e to the negative x, what do we do from that point? Multiply by x squared. So here's what we don't need to do, which is kind of typical to what we do need to do on some other problems. This is not going to be correct. We don't need to say, OK, now we failed with this linear expression times e to the negative x. Let's come up with some arbitrary quadratic. Well, we don't need this, and we don't need this. We've already ruled those out. Those don't help us. Those are pieces to the homogeneous solution. So we could go ax squared e to the negative x. If I remember this problem correctly, that also does not work. And that's really frustrating, kind of annoying, borderline annoying. Now, it actually crosses over. It is annoying. So this doesn't work. You can't generate x e to the negative x with that as well, even though it's not part of the particular solution. So this kind of came up the other day in class, too. If we determined that this, and some of you had success on this problem, you determined that this was not the particular solution, how would you determine that? Make them. Say that again, Jacob. Maple. Maple? Oh, OK. Well, that could be a determining factor if you had accessibility to maple, which we don't have on a test. I can try to do it with something weird, like 2a equals 1 and 2a equals 0 or something. I remember that trying to plug this in, this is kind of bizarre, and we do not get a solution for a. So we rule that out after some effort. You do this, and you do this, and you plug them into the original equation, and you're going to try to generate and 1 x e to the negative x, and it's impossible to do that with this problem, with this particular solution. So we scrap that one as well, and then we go on to a x cubed e to the negative x. And from what I remember about this problem, you can successfully get a solution for a, starting with this as your particular solution. Yes, product rule. So there's the first term of your product, there's the second term of your product. Would you have to, because you didn't really determine that a x squared e to the negative x wasn't, did we determine that that wasn't part of the solution, like would you not have to make it a quadratic, including? Like I'm sorry, like in the first part we did, we crossed out in the second try, we didn't use, when we did a x squared plus b x plus c, we didn't use b x plus c because they're part of the homogenous one, they'd be equal. Okay, so your question is do we need to include the quadratic in this one? Yeah. The quadratic did something bizarre, I can't remember what happened, but it, it created a problem and it very well might be part of the homogenous solution. We only found two terms that caused this to be zero with this shortcut, there may in fact be more, in fact maybe this is one of them. So I think once we rule it out, we don't need it anymore. So when we realized that it failed here, the quadratic term failed, we can just go on up and as it says in the recommendation, if I can find it, that it says if you fail with one, keep multiplying it by x, modification, I think they call it a modification principle or whatever in the reading, but when we fail with this, when we multiply it by x and try again, yeah, that does work, a x cubed e to the negative x does work in, when you do the problem, you can find a value for a, so you're not putting in a into your answer, you'll know a number for a that works in this problem. So hopefully that'll jump start you on that problem if that would cause some difficulty. The other problem that was recommended that we take a brief look at, so we have a little extra information at the end, what purpose does this serve, why would, what additionally are we gonna be able to do by the fact that we know the y of zero is one or the y prime of zero is zero? Find our c values. Our c values from the. Homogeneous. From the homogeneous part of the solution. Is this one where they gave you the complementary part or the homogeneous part? Okay, so r squared plus r minus two, characteristic equation. So we need factors of negative two that'll give us plus one in the middle, plus two minus one, does that work? So c one e to the negative two x plus c two e to the x. The c values now. Okay, that's a good question. I was gonna ask that, but since, that I decided not to, but since you did, let's go ahead and. Finding the c values now, so should we bring this into play and this into play now? Because that is the final y, that's not just y sub h of zero equals one. So that's the final y of zero equals one. So we've got some work to do. If this were a homogeneous equation, then we can opt right now to solve for c one and c two, but we've got some other work to do before we do that, which does affect the y of zero being one potentially. You could do this in pieces or you could do this all at once. The only way you're gonna be able to generate an x term on the right side is to start with what? x plus b. Do you see any reason why we would have to kind of get rid of any part of that solution based on what we have thus far in the homogeneous solution? Don't see any. Double roots or duplicity of roots or anything, so I think that should work. We should be able to go through and find out a and b so that we could generate one x and then the second piece, I'm just trying to get it to the point where you feel like you could take it over. Now we've gotta generate one sine of two x. How could we generate one sine of two x? Okay, give me some more other than that. Right, we don't know how many sines to start with and we don't know how many cosines but you've got the right terms. We need a sine, let's say c cosine two x plus d sine two x and then you can go through and that's the only way you're gonna generate that kind of term is with one or the other of those. Do we probably need them both? What happened in a problem Wednesday in here? One of them can't take that. Yeah, we didn't really need them both because we only had the original function and its second derivative which were the same, right? If you start with a sine and you take the derivative twice, aren't you back to a sine? We need them both here because we've got the original function and the first derivative and the second derivative. We're hoping to keep the sines involved in the problem but we eventually want the cosines to drop out but we need to start with both of them. For p one, when you take the second derivative of that, it's just zero so is that supposed to happen? But the issue is can we generate one x by plugging in something like this for the particular solution and yes we can with that term because the y term has an x in it. The derivative term doesn't have an x in it but that's okay, it's present here. So any issues about what to do to find a and b from this point?  Everybody okay with that? Take the first derivative, second derivative, plug it into here and try to generate one x. See what a and b have to be in order for that to happen. Start again, find the first derivative, second derivative with the second piece of the particular solution, plug them in here on the left side and try to generate one sine of two x and solve for c and d in order to do that. This kind of lengthy problem because you're not done at that point in time. Once you find a and b, if in fact they're both present which I know a is, I don't know if b is or not. And c and d, then you come up with the final solution. Does anybody happen to have the a and b and c and d values for this problem? Andy? So this won't be enough, this will be a number. I'm gonna have an a here but you'll have a number by this point in time in the problem. You'll have a number there. I used a c, I probably shouldn't have. So let's call this d, you'll have a number there and you'll have a number there. So the only unknowns at this point in time in the problem are c one and c two. Some of these might be gone. I mean it's possible that I don't, I think you're gonna need them both but it is possible that d or e could be zero. Then you would need to bring these into play, right? Take the original equation for y plug in one every time you see an x plug in a zero and then see what value for c one or c two would give you a y of zero equals one. Probably gonna have an equation with c one and c two both in it, right? And then take the derivative of this long mess that we have now plug in zero for x and it should kick out zero for y prime which will be on this side. So I think you're gonna have an equation with c one and c two here and equation with c one and c two here. Put those two equations with the two unknowns together substitution, linear combination, something to get the two solution. That's a lengthy problem. Probably an unfair test question. It just takes too long. That would be the only question on the test then it'd probably be fair but that's probably, chances of that happening are, well there's two chances of that happening. Slim and not, okay? So you're gonna have more than one problem on the test, Nicole. I told P one. Yes. But you just said I, I just. You need the derivative to plug in there which is what? A, you need the second derivative to plug in up there which is zero. So when you plug those things in, let's go ahead and do that. Second derivative is zero plus the first derivative which is a, sorry, minus two of the original y things and we wanna somehow figure out what it takes to generate the x term, the linear term. We're not gonna generate the sign of two x with this stuff. So when you equate the coefficient of x on the left side which is gonna be what? What's coefficient of x on the left side? Negative two a, it better be equal to one. So that's the only x term on this side. Everything else would be a constant. So a and what, minus two b. So the coefficient of x here better be equal to the coefficient of x here and this constant on the left side better equal the constant on the right side while the constant on the right side is zero. So a is what, negative one half which makes b what? Negative one fourth. So back to what I wrote out earlier. Once you write out the particular solution I wrote out a, x plus b.  A is negative a half. B is negative a fourth.  Like the values that you found for a. Yes. So the first piece of the particular solution instead of just being a, x plus b is negative a half x minus one fourth. Now you go on to the second particular solution and find, I wrote it c and d probably would have been better written d and e. If you have your supplement with you or the copy that I made of the supplement there are a couple of diagrams that I think are helpful to look at as we go into section 7.9. Not a big deal is made about some things on the picture but I think we wanna point out some things. Now it seems awkward that the page number is 1,172 but the first page on 7.9 you'll see a couple diagrams and let me back up a half a step. So we're gonna do some applications now some of which we've actually kind of started a little bit when we looked at how it is we might need to use or apply second order differential equations one of which was a spring. We talked about the Hooke's law and the restoring force and mass times acceleration those kinds of things. They've already been mentioned in here one time but the diagrams that you see on the left side of the page you see the spring with the mass here at the bottom of the spring. Notice the scale out to the right that if this is in equilibrium position it shouldn't be a mystery that the equilibrium position is zero but look what happens when you go down the scale. It's a subtle way of telling you that in this particular model down is positive, this diagram. So as we go let's say we've got this mass at the end of the spring it sits here at its equilibrium position in order to start the mass in motion let's say we pull the mass down and release it well we need to prepare ourselves for the fact that pulling the mass down and releasing it down is positive on this particular picture. Now the other picture it's not a stretch for you to think of we've got a spring here they've got a scale down here on the other direction where the equilibrium position is here and they move it out to the right and it's not a stretch probably shouldn't use that word it's not a leap to think that moving it out to the right beyond its equilibrium is positive that seems positive but this one doesn't really seem positive down we normally think as negative but it's positive here that kind of starts the oscillation and you say in this particular model if there isn't any friction which we start the first model we look at doesn't have any friction at all I know that's not good but then we'll see what happens when we throw friction into the picture. So if you take this mass that's at the bottom of the spring and you pull it down positive beyond equilibrium and then you release it it's gonna oscillate so this spring the position of the spring over time it's gonna go back up beyond its equilibrium and then come back down and if there isn't any friction it's gonna oscillate indefinitely so that makes sense this spring once we set it in motion is just gonna stay in motion forever there's no friction to stop it so it looks a little bit redundant when you say it goes back up here comes back down to here well let's get a picture that kinda scrolls for time so if we let T be along this axis where is it? What's it do? Well we started it up here as you scroll time it comes back down here and then it goes back up to where it was and then it comes back down here and then it goes back up to where it was so we've got this nice little oscillatory model in reality it's going kind of moving in a straight line but if you let time scroll along the axis and show how it moves what would you say that type of oscillatory model is? Sine, cosine or sine plus cosine? Because if it is a sine plus a cosine we could, and we've already done this once we could put that in the form of a sine only or a cosine only and this might be a good time to take an answer that we had earlier in this class four or five days ago we graphed this, I'm not gonna graph the whole thing again but we graphed the sine, we graphed the cosine we added them up on the coordinate plane and we ended up with a function that itself was oscillatory we went back and revisited that and we gave it an amplitude of square root of two square root of two, is that ringing a bell? The final graph that was sine plus cosine we went back and revisited that another day and we said if the period of this is two pi and the period of this is two pi when you add them together they have the same period which was also two pi so we left the coefficient of x one and then we figured out kind of backed it up a little bit and figured out what the phase shift was I'm not getting any good visual prompts from you is this ringing a distant bell? Yes, okay and what was that phase shift to the left? Good, okay. Thought maybe I was talking Spanish there for a while I do that from time to time so this sine graph with a changed amplitude up the square root of two units down the square root of two units period of two pi and a phase shift of pi over four to the left is exactly the same graph as sine x plus cosine x. I wasn't gonna do this but I think it might actually be worthwhile to do this. We drew pictures and I don't have the pictures here right now so since we don't have the picture let's see if you remember this it might not be a fair question. The sine of the sum of two things. What is that? Sine of... How is that? Does that look familiar? I guess it was a fair question. Thank you. Now let's, what are our two, what are our a and b values? Well a is just x we're not gonna be able to do much with that but b is pi over four so let's see what we get from the sine of x plus pi over four. So it should be the sine of the first cosine of the second plus cosine of the first sine of the second. Everybody okay? That this equation kind of morphs into this equation with this sine of a sum identity. What's an exact value for the cosine of pi over four? What's an exact value for the sine of pi over four? Same thing, right? Sine of pi over four and cosine of pi over four are the same. All right, let's distribute the square root of two to both terms. So what's the square root of two times the sine of x times the square root of two over two? Sine x, is that right? What's the square root of two times the cosine of x times the square root of two over two? Just cosine of x. Is that what we started with? Sine x plus cosine of x was our original. So they are the same. We're not validating that by a picture. We're validating that kind of algebraically with some trig identities thrown in. So you can take this thing that is sum of sines and cosines and write it as a single sine function. This is called a phase amplitude form. In your supplement, on the top of page 1173, you will see kind of some conversion, some interrelationships between something given to us in this form and how to convert that into the so-called phase amplitude form. I'm gonna actually give you something slightly different from this that I think simplifies it a little better. So that if we have an answer that is the sum of sines and cosines, so this is a unit circle and it's a unit circle centered at the origin, so the equation is x squared plus y squared equals one. Here is a point on that unit circle. This angle phi or phi is actually this angle right here. So it's from the zero gradient or zero degree mark rotated through positive phi or phi radians which takes us to this place in the plane. So here is a point on the unit circle. Typically when you see a point on the unit circle the x value I'm hoping that some of this rings about because we haven't really looked at any of this in this course, the x value is the cosine of the angle that brought you to that point. The y value is the sine of the angle that brought you to that place in the plane. So from this angle phi or phi we have the cosine of phi is a over the square root of a squared plus b squared. The sine of phi is b over the square root of a squared plus b squared. So those things are stated right down here. Do you remember that from a unit circle that the x value is the cosine of what got you there and the y value is the sine of what got you there? I guess we could validate that by saying the x value is here and the y value is here and the hypotenuse is one right and you could say sine is opposite over hypotenuse cosine is adjacent over hypotenuse and you could come up with these identities. If that's distant for you then mess with that a little bit till you feel assured that this is legitimate information. So far we haven't deviated a whole lot. I guess we're going to deviate from this one that's in the supplement because they convert it to a cosine and we're going to convert it to a sine. Maybe one more thing before we completely leave this diagram alone. This is supposed to be true. If you plug in the x value you get the cosine of the angle cosine squared plug in the y value you get sine squared certainly cosine squared plus sine squared is equal to one. You could also plug in the coordinates if you take the cosine or in this case the x value and square it plug in the sine value and square it or call it the y value that's the same thing. You can check that out to verify that you do in fact get one when you square those and add them together.  This is going to look familiar but we had specific numbers earlier. I'm going to propose that the amplitude in the phase amplitude form is going to be the square root of a squared plus b squared. Let me see in the what do they call this? Yeah they do call it an omega a lower case omega so I'm going to use the same looks like a w in this phase shift angle the angle that we're talking about on the unit circle diagram is the angle phi p h i so that is let's say that's what now we had this in terms of x this particular example has it in terms of t same thing if we were to use that sign of a sum again it would be sign of the first cosine of the second plus cosine of the first sign of the second from the unit circle what was the cosine of phi it was the x value of that point which was what and what was the sign the sign was the y coordinate of that point on the unit circle if we distributed this thing out in front like we did before it was a number it was the square root of two now it's the square root of a squared plus p squared what is the first term would there be an a left? because the numerator and denominator would knock out so you'd have a sign of omega or w t and if you do that same thing to the second term don't you have a b cosine of omega times t isn't that the form that we're gonna have when something oscillates right? sign plus cosine can we take something in this form this is kind of in reverse but hopefully it justifies if it works this way coming down this page couldn't we start here and work our way back up to this equation if it's an equation as we work down the page we should be able to basically say they're equations also as we work up the page so we're gonna have something like this and we want to put it into this form so how do we find the amplitude? well we're gonna have a number here potentially and we're gonna have a number here so we are in which we haven't done this yet we picked it off of a diagram didn't we? we added the square root of two over two to the square root of two over two and we got the square root of two on the diagram now we know with numbers whatever the numbers happen to be we'll take a which is the coefficient of sign and b which is the coefficient of cosine we'll square them and add them together take the square root that's gonna be our amplitude once we put it in this form let's check it we started with this one should the amplitude have been the square root of two? yes one here and one here one squared added to one squared is two square root of two that would be the amplitude uh... I don't want to skip over this this got mentioned but let's the omega value is here and here it will also be the same up here we want the periods of the individual sign and cosine functions to be the same as the period of the so-called phase amplitude form we want them to all have the same period because they're gonna all repeat in the same cycle so omega is unchanged from the original sine plus cosine equation to the phase amplitude version how do we get the angle phi see if this makes sense from our diagram back to the unit circle diagram if the x value is the cosine and the y value is the sine or if you just wanted to use the right triangle here tangent should be what sine over cosine so if we put this thing over this thing what would you get b over a so if you want to find the so-called uh... phase shift or the phase angle then take b now look at what we have for b actually doesn't that look really good no it's not on that page so on this page what is b b is the coefficient of cosine sometimes we write that first we write the cosine term first and the sine term second so just remember that b the coefficient of the cosine term is in the numerator a is the coefficient of the sine term that's in the denominator and if we do an inverse tangent problem then we can get the angle back to our example we had sine of x plus cosine of x if we did tangent of phi we would get b over a which they're both one coefficient of cosine is one coefficient of sine is one what angle has one for its tangent five or four right and that's not what we got this way but we got it another way so this way we don't have to draw them figure out what we think is the the phase shift involved in that graph we can do this inverse tangent problem so in our example we got pi over four so is it possible without drawing a diagram and i'm gonna change to x is now to go directly from this version to a single sine or cosine we've got sine so what's the amplitude the amplitude is the square root don't need a graph for that here's a one here's a one square add them together and take the square root square root of two sine well the coefficient of x here is one so our omega or w value is one same thing here so we're gonna have one x again tangent of this angle is the b value over the a value one over one what angle has one for its tangent pi over four now by adding pi over four the net effect is to shift the graph what? left so it's actually going left pi over four i think we found that to be true on our picture which we brought over here and saw that it actually was shifted by pi over four units to the left so you can go from this version to this version without graphing it and trying to figure out from the graph what's the phase shift what's the amplitude the periods will always be the same where did you get the x? this is this one? the x in here whatever the coefficient of x is in the sine and the cosine it's going to be the same here because the period of this sum it's got to be the period of the net single sine or single cosine graph so that's slightly different from what's in your book but i think getting the amplitude there and getting the phase angle here is going to be a little more straightforward than what is illustrated in the in the book itself alright let's set up a problem and that's probably all we're going to get is the setup alright thank you hook's law which we have already encountered a couple of times in this class says that the force required to stretch or compress the spring is directly proportional to the elongation of the spring itself how long how far it's already been stretched or compressed so we've also used this at the restoring force then if the force required to stretch or compress it is k times x that's hook's law directly proportional to how long it's been stretched the restoring force is that opposite motion so the restoring force is going to be mass times acceleration and the forces involved in this motion of this spring with a mass at the end of it they've got to be equal so the restoring force basically given to us by hook's law and force being mass times acceleration these forces have to be equal you put everything on the same side you get this and if you want to make that look a little more familiar to what we've been doing in this material m is the mass it's a number second derivative of x which is really just acceleration but second derivative of x we'll call x double prime isn't this a second-order linear homogeneous differential equation it is there's nothing in this problem about friction or any kind of a damping force at all when that is thrown into the equation so here we've got a second-order differential equation when we throw in the uh... damping force or the friction into the problem we're going to have this and then we'll get this written down and then we'll stop at that point uh... i don't want to write it like this first time let me scrap that so here's our mass times acceleration we've got our kind of hooks law restoring force over here now let me read this next we consider the motion of a spring that is subject to a frictional force in the case of the horizontal spring or a damping force in the case where a vertical spring moves through a fluid an example of the damping force supplied by a shock by a shock absorber in a car or a bicycle okay here's the statement that's going to get us what we need out of this equation we assume that the damping force is proportional to the velocity of the mass and acts in the direction opposite to the motion velocity well if this is acceleration the second derivative of x what's velocity first derivative of x so there's our damping force there's our friction so now this is what i was going to write a minute ago but i wanted to write it in this form first it'd be nice if i could actually write it i want to bring these terms over to this side x prime is really dx dt now we have a second order linear homogeneous equation but it has all three terms there's our second derivative term there's our first derivative term there's our original x term and we are equal to zero we'll pick up at that point