We're very excited to have you here today. This workshop today is about Red Hat OpenShift API Management, which is our API Management Service. So I'm here, your presenter. My name is Jennifer Vargas. I'm a Principal for Marketing at Red Hat. And with me today here is Jaya Paskaran, who's our Technical Marketing Manager.  So let's get started. So to get started, I just kind of wanted to do a quick product introduction so you know where we are, what the product is. Talk a little bit about the workshop details and share the links on the screen. I already shared with you the links on the comments tab. So go there, check it out, copy the links. There's two of them. And let's get started. So Red Hat OpenShift API Management, it's part of a larger initiative within Red Hat, which is our Red Hat Cloud Services. The Red Hat Cloud Services is a set of new cloud services that we are bringing to the market. And it consists on three types of services, the platform services, application, and data services. All of them are natively integrated with each other. And the idea of these services is to make sure that we're providing you a unified platform to build cloud-native applications. And OpenShift API Management, of course, is a key part of this, as it can help you today on managing your APIs and making sure that when you create applications that are API-driven, you know where you're saving all your configuration, you know where you're managing your policies in terms of who gets access to those APIs. So as I just mentioned, Red Hat OpenShift API Management is a fully managed service for API management, of course, sorry, this sounds like a tongue twister there. But it's delivered as an add-on to manage OpenShift. So if you have an OpenShift dedicated cluster, or you happen to have Red Hat OpenShift in top of AWS, which are our two offerings on AWS, you could get API management running on it today. Actually, we do have a free tier. So if you're an existing OpenShift dedicated customer, you could actually get this running almost in a few hours. So what we have on API Management is a couple, the components are here on the left side of the screen. So the first one is the API Manager that allows you to see all your policies and it allows you to define them and check out all your APIs. On the second one is the API Gateway, which basically is the one that enforces those policies and enforces the API governance. And then finally, you have the single sign-on capability. And what that does is it helps you out to secure your APIs. And all of this is delivered on top of OpenShift and we provide you with 24.7 premium support and a 99.95% SLA. And one of the things that we have done with these products to make sure that since it's natively integrated with OpenShift, it provides you with a streamlined developer experience that you can interact very easily. So it is a great platform for allowing you to have an open, modern approach to building API-first applications. So I mentioned this already. We do have a free tier available. So just keep an eye out there.  So what do we have for today? I'll leave this for the end if we have a chance. For today, one of the things that we want you to do is to instruct you on how to use OpenShift API management. And the way we're going to do that is using something that's called the OpenShift API Management Sandbox. So the guidelines for how to get to those pages are on those documents that I just shared. So the goals for the session today is first you're going to provision your dedicated sandbox environment and what that's going to give you is access to an OpenShift dedicated cluster. And within that cluster, you're going to be able to access the OpenShift API Management UI. And within the OpenShift API Management UI, you will be able or well with the OpenShift cluster, you are going to be able to deploy a containerized application using some of those OpenShift console capabilities. And once you have that application in there using API management, you're going to be able to secure and manage your API. The team or Jaja is going to make sure that you find all those quick, nice ways to understand how to use the product. So what do you need today? It's very simple. So you need to create a Red Hat account and have your credentials handy. This doesn't take very long. And you'll get access to the sandbox. There are a couple of sandbox within our developers redhat.com page. Today we are going to put all our efforts and our attention towards the OpenShift API Management Sandbox. Go ahead request that sandbox and we will walk you through the step-by-step instructions for today. Links, very important. Here's the link. If you didn't get it on the chat, you can write this one. It's very easy red.ht slash api-workshop. That will get you to the guide in the guide. You have a little bit more information about API management, a couple of useful links. And it will take you to our scholars page. The scholar page will have all the instructions. Very well easy to follow, numbered, and you will get everything that you get in there. You can actually do this by yourself. Probably you could pay attention to us today and then maybe come back tomorrow and do it from scratch. The guides are going to stay live. The developer sandbox is available for you all the time. You get access for 30 days to the sandbox so you can come back whenever you want to try it again. And once that sandbox expires, you can just request another one. So you can have a lot of fun with API management if that's your choice. So let's get started. So I'm going to pass it to Dalia. At this point, I think we are good on the presentation side. Let me stop sharing here. Hey, Dalia. Thank you. Thank you, Jennifer. Good luck on your presentation. I hope everything is good. And if you want me to be here, I can come back at any time. We can talk more about the sandbox or any questions that our wonderful participants have. Hello, everybody. Thank you so much for joining us today. I'm just going to leave the link again once again for you to be able to access that. What you would be able to see is this page. It's a Word document, it's a Google document. You would be able to see all of those instructions here. And the first step is to create a sandbox. Now, what is a sandbox? A sandbox is a sandbox, like children play, typically. So there's an environment for you to play and there's the developer sandbox, specifically for OpenShift API management. You can manage and secure your APIs. You can try out, like Jennifer said, you can use this environment to try this out for 30 days. And then you can again register for another one, very soon, again, if you want to do this again. All right. So I hope some of you have already created some of the user ID. If you have not, we will go ahead and do this. I request you to go down to the last step. It's workshop, step by step. And there's a link right over there. Please click on that. And this should take you to the Red Hat OpenShift API management workshop. I hope you're all there. If somebody could say a plus one in the chat, then I would know that you're able to access this page. I'll pause for like 10 seconds. Okay. I'm hoping you're able to see this. And this is just an introduction in terms of what this workshop would, what you will typically do in this workshop. And clicking on the setup, the developer sandbox. And in this case, please work, please sit down. Let us do this together in case you haven't done this. Click on the Red Hat OpenShift API management user activities page. And this should open this up in a separate tab. And click on launch your developer sandbox for Red Hat OpenShift API management. Like Jennifer said, this is definitely a tongue twister. If you have a user ID, please go ahead and register. Login already. If you don't have, click on register for a Red Hat account. Create a user. As you can see, I have done this multiple times. So let us assume definition. I hope you are with me. You can do the exact same thing if you don't have a user name already. This is the regular terms and conditions. You're welcome to have a look at this or just accept this because it's a regular terms and conditions that you would have seen again and again across Red Hat. Click on create my account. You would be taken to another page where we ask to launch it. We come back to the same page while the API management is being provisioned in the background. It may kind of take a few seconds and then your account should be ready. Now start using your developer sandbox. Go ahead and click on that. You'll be asked to provide a few more additional details. I hope you have your email address handy because your email is handy because you will have to approve this portion of that. Now this step might ask you to provide your phone number just to validate using an OTP. Rest assured, we are not going to save your phone number anywhere. It is purely to protect our environments from mining. So because I have user Red Hat email address, you would not seek that step, but if you are using, of course, most of you might be using your own email address, right? So you may have to go through that step. So with this, I finished my setup. I'll be receiving an email. Once you've reached this step, you will be receiving an email, which would ask you to verify your email address. And once we do that, we will be able to see the Red Hat OpenShift dedicated page, which is basically the sandbox. Now, the Red Hat OpenShift API management is the sandbox is deployed as a tenant over here. And for the OpenShift API management to be able to access the services which will be deployed on the OpenShift dedicated itself, we will have to set up certain network policies. Now, typically, this would be automated, but at this instance, you will have to create one network policy if this is already not being created. Yeah, so go ahead and pick the dev project right from the top button, and you should be able to see a page like this. The network policies page, please ensure that you have picked the dev project. So this would be your username hyphen dev. Like every other sandbox, you would have a dev as well as a stage, two projects automatically created for yourself, two namespaces automatically created. Please use the dev namespace. Now go ahead and click on the create network policy. And we will directly go and change the YAML. So once you're in this page, click on edit YAML. And within the edit YAML page, you would be able to see this section where we have the content, this section of this, right? So please copy this portion of this. And this is basically, if you're, I mean, just very quickly showing this page again, under setup, under section one, please scroll down a little bit. And you would be able to see the YAML section that you will have to copy. So this is kind of a temporary arrangement. This is being automated by the engineering team. So this should roll out very soon, but just to ensure that we're able to progress with the workshop, this is like a workaround at this point in time. Then this enhancement is getting coded as I speak. So copy this and then go over to the YAML file and everything from spec converse, right? Just replace the whole thing. And the alignment sheet should be fine. I mean, it should automatically be a pre-aligned. And click on create. And you have created the network policy. Basically, if you see this, what we are saying is that we are allowing access from an ingress perspective for the ROM sandbox. ROM is that I have OpenShift API management. It's like an acronym. So this network policy allows the OpenShift API management to speak to OpenShift dedicated, which is a sandbox that you see over here. Now, once this is set up, we can go ahead and create the tenant. So switch over to the developer perspective. Now, we have two perspectives. If you are not familiar with OpenShift, you have an administrator perspective and you have a developer perspective. Switch over to the developer perspective and we'll go ahead and add a container image, an existing image. And then click on the container images. What we're trying to do is that we are trying to we'll deploy a pre-existing OpenShift, pre-existing quad-price image over here, which is based on APIs, which would be the API that we would look at managing and securing. So choose this external registry or pick up this API. It is all the instructions are here. If we click on the deployer as API, you would see that the image name here. So click on the image name and choose that over here and just for one sake, let's pick up the right icon, pick up quad-price as the runtime icon because it's a quad-price image.  You don't have to change anything else here. Click on create. Now, what this does is picks up, downloads the container image and deploys the application for you. It should take just like a few seconds for the application itself to get deployed. When you see the dark blue ring over here, which means that this particular image has been deployed and clicking on this icon, external link icon, you would be able to open up and see the image. That's the welcome page of quad-price. Now, this open, this quad-price application has been annotated of to already have to be able to expose the open API specs. And let us go ahead and see how that looks. And we'll have to ensure that we choose HTTP instead of HTTPS here. And you would be able to see the API, open API specs version 3.3. Okay. So we know that this application has been deployed. The open API spec is working as it should. And let us go ahead and do the next step of what we call service discovery. Now, service discovery is a feature that helps you to import services from an OpenShift environment, OpenShift cluster, on to that had OpenShift API management. And this is automatically possible when the OpenShift API management is in the same cluster as where you have the service deployed. In this exercise, we will look into that option. You would have other resources and learning materials where you can deploy APIs or manage APIs, which are not in the same cluster. But in this case, the OpenShift API management tenant is also going to be in the same cluster. Now, I have deployed the application, but we also have to I have skipped a step here, my apologies, of actually creating the tenant itself. So click on search and if you click on the resources, you will be able to see the API management tenant. When I click on API management tenant, it asks you to create the API management tenant. Go ahead and click on create button right at the bottom. And this YAML file, we just go wait for a few seconds for the YAML file to, sorry, for the tenant to be completely deployed. So when the tenant gets deployed for all of the stuff, the magic which has to happen in the background, let us go ahead and annotate the service that we have. What do we mean by annotation is that, like I said, for service discovery, the feature which helps you to import the APIs easily from an OpenShift cluster into the OpenShift API management system. What we will have to do is that we have to tell this API that you have to set it up for auto discovery. So click on the services, right? This is the services we are going to annotate the service. Click on Rome Quarkus Open API services and here is where the annotation happens. Let us go and have a look at how this annotation happens. So the first one is that we will have to say that it shouldn't be discoverable on the answers, yes. So this is the label that you will have to add to the service. In addition to it, we also add a few other annotations. The one is what is going to be the API path, the description path and what is a port and what is a scheme, right? We have to look at these three things. So let us go ahead and annotate the service. The first section is a copy over on this particular label and let us add the label to this particular Open API service. All right, let's stop checking, click on save. The OpenShift, the discovery label is set. I will also have to add the additional annotations. The first annotation is the Open API path. The next one is the port. The port is 8080. The last one is the scheme. That's a space. So you should have four annotations here and the label that we had created. So with this, we have set up the service discovery aspects of the Quarkus API and you are able to now discover this from the Red Hat OpenShift API management system. Let us go back and check. The way we can check easily is to click on the project, the left-hand side, and you can see the API management launcher link over here. So we go ahead and click on the OpenShift API management launcher. That will open up the OpenShift API management, the ten-inch that we had installed. OpenShift API management is based on three-skill API management and that is why you see that icon over here. Now, you will have to authenticate yourself using the Red Hat single sign-on. This is important. Please click on this link right at the bottom. Authenticate through Red Hat single sign-on. All of these instructions is over here with enough pictures so you can either follow along by spinning up a workshop or just listening at this point in time and try this out later. Then you go ahead and log in with Dev Sandbox. Now, this will take you to the dashboard of OpenShift API management.  This is how we create a product. A set of APIs, a set of backends can be created together to form one single product. Let us go ahead and create the API, the product tagging. Click on Create Product. I'm hoping all of the magic has already happened with the service discovery and we should be able to import it from OpenShift. Now, this is again, we have to authenticate this to enable this option. So please click on that authenticate link and log in with Sandbox, Dev Sandbox again, to authenticate that had OpenShift API management. Go ahead and click on Import from OpenShift and you can see that there's one single product. We had this one product called Rome Quarkus Open API and then within the namespace Dev. So it automatically discovered those APIs and because there's only one of them, only one product over here, because we had just one API, one service deployed, right? That's what appears over here and we click on Create Product.  The product itself gets created. Like I said, an API in OpenShift API management is referred to as a product and each product can have more than one backend, one or more backends. And what is a backend? A backend is typically a web service on that OpenShift API management with proxy requests based on certain rules that you define against each product. So the product has been created. Let us have a look at how this product looks like. It's got the name, the system name and there is a backend which has already been created. Now, if you have a look at this link to that backend, let us very quickly have a look at this backend. It's got a private base URL. Now, if I go back to where I had deployed the application on OpenShift itself, if I click on the icon, I would be able to see the service. Now, the service, under service routing, you would be able to see the host name. And this is the same host name that you would be able to see on this page, right? It is the same service that is being used as a private base URL. This works because you are in the same cluster, but if you are in a different cluster, you would be using a different URL, which would be like, you know, internet-facing, the route, perhaps, not necessarily the service URL. But in this case, like I said, it was auto-discovered and so the backend got created. Now, let us go back to the products. We go to the products menu. You'll be able to see a listing of all the products, which are available here. I click on the product that we created just now. You would be able to see the active docs. And I'm not able to see the active docs. So, here it is. Give me a second. I'm just going to double-check if my network policy is working as it should. Just give me a second. And this is basically just to show you that there's a real demo, not a recorded demo, that we are having such a few challenges. I'm just going to double-check if the network policy that we were supposed to set up is working as it should. I'm just going to try to create this network policy one on third time and say, allow me to go back and ensure my policy is correct. And then I'm just going to check if my APN tenant is functional as it should. Okay. And I'm just going to try to import this once more. I'm just redoing the whole thing again. Now, hoping the new network policy kicks in. And oops, I have to go ahead and set up all of those services. This is like a revision for all of us. Now, to ensure that we know, understand, remember everything that we have to do thoroughly. So, I'm just going to go and annotate the service once more, the service is annotated. And add the rest of the stuff right. All of the annotations really go ahead and do the annotations once more. Nothing has crossed that it's been annotated. Now, I should be able to go and discover it once again. Now, like I said, we have a new service with a different name type. So, you'll be able to see that. Let us go over here. Let's check if the active docs are appearing. Yes, I think I had made a mistake and now the way I had created the network policies are my apologies. All right. So, the active docs has been created. If you click on the active docs, we open this up. This was the same spec that you had already seen. I had opened up the active docs. Let's give it a few seconds. Yeah. And you'll be able to see the active docs all of those, there's an endpoint called fruits that you are able to see. And let me go back in to the service. If I click on this, the URL, the one I had deployed, it basically shows you a JSON file of winter fruit and a tropical fruit, an apple and a pineapple. So, these are the various aspects, these are the various methods which are available in this particular generated API. Now, the next step is we will have to, now you may ask me now, fine, you have imported what next. What we will have to do now is to go ahead and secure this API. Click on integration and configuration and what you would be able to see here is the staging APCAST. Okay. So, what is an APCAST? Open ship API management routes, all of the API requests through proxies, okay, known as APCAST instances. An APCAST instance is an NJNX based API gateway which helps you to integrate your internal and external API services with OpenShift API management. It also helps you to enforce your security policies and rules and now we will see how we can actually go ahead and set this up. Now, if you can see over here, right, we have a call command saying that there's an example called for testing, but the user key is just like a temporary place hold on. So, if you look at the instructions right at the bottom of the page, it says that you will have to start by creating an API and application plan. Now, you would be hearing a lot of such terms, right? We spoke about product, we spoke about backend, then we spoke about APCAST. Now, we also have another term called application plan. In OpenShift API management, an application plan is used to define how the APIs should be used, what kind of usage rules, what kind of limits should be applied on the API. So, when a developer signs up for your API, they can select a given plan. So, you may have different plans, you may have free tier, you may have paid plans, you may have gold tier or platinum tier, enterprise tier, multiple tiers, right? So, that is where you would be creating different application plans and the developers would sign up for a particular application plan, depending upon their need. Click on start by creating an application plan and let us create a new application plan. Now, we are under the same service, OpenShift API 1, and I'm just going to give certain names over here and click on create application plan. The plan is created now and once the plan is created, you will have to publish the plan. Till then it is in a hidden state, okay? So, I have published the plan. Now that I have published the plan, I will have to go ahead and allow a developer to access these plans, right? And that is called an application. So, you will need to create an application which is associated to a developer. Now, all the developers or accounts are under the audience section. If you click on the audience section, you would be able to see a list of accounts which are already there. You can create your own account or for demo purposes, we can go ahead and use the same developer account which is automatically pre-created for us. And click on the developer account and as part of OpenShift API management, if you remember right on the front page, there was one application which is already live, but we wanted to, we would like the developer to access the Open API, the Quarkus API that we had created. So, we'll create the new application and as part of creating the new application, we'll have to pick the right product. Now, there's the API one is the right product and what is the plan? There's the plan that we had created and we just pick our name and the description. I'm sure in real life we wouldn't be calling it this. We may be calling it a better name, but let's just go with what we have here. Create the application. Now, when you create an application, you would be able to see the user key and of course you can go ahead and reach under the user key and whatnot. So now, this particular user will use this user key to be able to access the API that we had deployed. Let us go back to the product and under integration, if you go back to configuration. Okay, and if you remember earlier, we had the user key deployed as user underscore key in caps. Now, this is an example call. Okay, so if we have multiple users who are using the same API account, you would just see one user key over here because this is a sample, this is not like the final thing. So, this is how typically one of them would use. A developer would access that. In this case, this is passed as a parameter. There are, I'll show you how you can, you could change this parameter to be a part of the header, perhaps, and access the API. Now, let me go ahead and try to access this without the user key. Okay, so you'll be able to see that. You know, it says authentication parameter is missing. And now I'll have to choose, let me use with the user key. And now you can see that application is accessible. Now, let us call the method fruits. And you can see that this particular URL has been, you know, it is accessible at this point in time. I'm just going to, I'll hit refresh a few times. Please go ahead and try doing that a few times so that because then we can have a look at the traffic pattern as well. In the meantime, what I would also like you to do is if you go back to the open Chef drive, the Sandbox where we had deployed the service, you can see the route. So this route that you see over here is also accessible. So if I click on fruits and use the HTTP instead of HTTPS, you are still able to access this. So it is not, I mean, if somebody gets a hold of this URL, they would still be able to access this. So if you could just go ahead and delete this route, okay, click on the route. And right over here on the actions, you will see a delete route. I'm just going ahead and deleting that route. When I go back to my topology, you will see the first API that I had created has got this open URL one, but this one does not have an open URL. It only has the service. So the only way you're able to access this API is using a service, which is the hosting that we saw already, right, says service cluster local. This is the only way you can access this. But as a developer, you'll be able to access this. So I'm just going to refresh this page again just to show you that you are still able to access it through the API. So what I'm saying is that from the APCAS, APCAS URL here, copy-paste, copy-use, use that URL, right? And you will be able to access that only through OpenShift API management and with the usage. Okay, now that we have accessed that particular page a number of times, let us see how the analytics is going to look like. You can click on the analytics page and analytics click on traffic. This is the quick view of it. You can explore this further, right? So it just tells me that I have tried to access this multiple times, 17 hits. So hits is this particular, hits is this, the hits traffic analytics is preset for you. So you'll be able to see the access of that. And for example, I had tried to access without the URL, right? So it tells me that there's no integration errors. But for example, if I provide all wrong user key, perhaps, right, authentication failed, then it would tell me that user key was invalid. And then it also tells you what are the kind of response codes that has been made available. That's a positive response code. And what does the daily average and so on and so forth. You'll be able to see primarily based on the hits, right? You can set up other methods and metrics later, but out of the box you have the hits metrics available. So that kind of brings us to the end of deploying your first API on OpenShift with a little bit of excitement where it wasn't working. And then we were able to import the OpenShift API, the service. We were able to deploy the Quarkus service on OpenShift. Then we were able to import that Quarkus API on the OpenShift using self-service discovery. And then we were able to set up the application plan and the application to ensure the user key is generated and we were able to access in a secure fashion. So we were able to deploy it on the sandbox and manage and secure your API. Now, OpenShift, as part of the OpenShift API management, we also have single sign-on, which is not part of the sandbox itself, but you could be able to use the sandbox. You can set up the sandbox, use the SSO, or it has single sign-on in the sandbox also. But before I show that to you quickly, I would also like to show you certain settings that are available over here. For example, on this point in time, this API cache is managed. And this is what I was talking about. So in this case, if you see the application API key, user key is identified and authenticated by a single string, as in how we saw it like a parameter. You could also use it like the HTTP header. If you choose HTTP header, you'll be able to... And for example, you want to call this as API key. Like you can just change a few of those settings. Let me go ahead and update the product. So any of the settings changes that we do will flag this up saying that there has been a configuration change and you will have to promote the new change to staging. So in this case, I have to promote the version 2 and then now if you see this, now it tells me the curl is this way. So the user key is no more user key. It is now called API key. And you can go ahead and run the curl command. You'd be able to see the same thing like what we saw earlier. Excuse me. All right. So I'm just going to pause and see if there are any questions. Thanks, Jonathan. I think I saw your comment a little too late. Perhaps take a second and see some of the settings correctly. Now let us just go back to the OpenShift dedicated place. So you would have two services, two projects that I spoke about. You have the Dev project where we set up the tenant and then we are imported or deploy the Quarkus API. And then we also have the stage environment. There are no services at this point in time. But like I said, you can use OpenShift. So I did have single sign on a container image over here. So you could try this out to click on developer catalog. I may click on SSO. You can choose anyone of the SSOs over here if you want to try out how to integrate SSO with OpenShift API management. Now the instructions for that will be available to you very soon. We will link the new instructions back to the same instructions that you have here. We will add additional links to further instructions as well. So you can, if you're looking to use single sign on as well as part of your API management, you can try this out as well. Just click on it in standard shape template and just go ahead and create that. That would create an SSO for you. It just takes a few minutes. So I would not wait for just to complete that. But that would be a workshop by itself. So that's what we have today. Are there any questions? Okay, I think so JWT is used when integrating with SSO. Yeah, OpenID Connect Auth is also available. Like I said, this sandbox is available to you for 30 days. But you're welcome to go ahead and spin up another environment. Again, another sandbox environment when the 30 day lapses. I'll look out for additional learning paths that we are deploying stuff like OpenID Connect. How do you do developers, developer environments? How do we do developer portal and so on and so forth? So we are definitely looking at launching all of those instructions as well. You should be able to see all of that soon. So here's where the developer portal is. And this is the default developer portal which comes up at this point in time, by default, obviously. And you will have instructions to how you can actually go ahead and set up new developer environments. Or how do you hit it this out for your own? So let us, if you look at this one to dummy user that has been set up for you. So this developer portal is available out of the box for you at this point in, which you can go ahead and hit it this out. So instructions for that would be made available to you later. And that's it for me, Jennifer. That's nothing else. I can stop sharing. Awesome, Jaja. Thank you, Cover, Bernard and Evan. Help, they are the other SMEs from our. We're excited that you guys were here. And it looks like to launch the guide and follow the instructions. This concludes our session for today. Thank you very much for joining us. Remember, just go back home and keep exploring the development is on there. That menu that has struck you to many other of our shift data science, like red hat open shift trains for Apache Kafka, which is our Kafka service there. And you can do them today. To stand up this environment as you just. Jaja, you did a wonderful job. Thank you, everyone.


Come on, everyone. Damn, here with late in sports cards. We've been two boxes of 22 tops, Formula One, Chrome Sapphire for Zach S. So good luck here, Zach. Let's pull something big, man. See a big variation. See a big numbered card. Yeah, no problem, Jeff. I'm pumped, dude. I haven't written the war in God. Maybe a 10 box Tuesday here or there, but like a personal box where noir is like on its own. I've not done that in a minute. So I'm pretty amped for that. Okay. Gonna be a yellow Chambuilik Bashee to 199. For Sheru, Charle Lecler, and Joe Guanyu. Dennis Houser, Sebastian Vettel, Sebastian Vettel, Sergio Pérez, Richard Bershore, there's a green Clement Novilock, Future Stars to 75 for MP. Nice one there on the green, Max Verstappen. Card number three is that variation? That Verstappen, 30. Now it is not. Never mind. Side of an Ato. Can be right, Reichenan, Total Wolf, Lance Stroll, Chambuilik Bashee, Oscar Piaustry, there's a yellow Kevin Magnuson for Haas to 199. Kevin Magnuson, Ryan Nassani, Fernando Alonso, Daniel Ricardo, Theo Porsche, Logan Sargent, Charle Lecler. Ooh, I think we got a variation coming up. That is gonna be, yep, card number five, Lewis Hamilton on the variation, I think. 52. I don't know, is that the variation? I can't tell, to be honest. I don't know if that's the variation, I don't know why it was upside down, if it's not. There's definitely a possibility it might not be. Stand by on that one. Verstappen, Grand Prix, Joe Guan Yu. Kevin Magnuson, there's a Sepia of Max Verstappen to 100 for Red Bull. Yuki Sunoda. Yeah, I think that is the variation, because I can't remember if fit five, two is the variation, I think, S-monoakon. So I think that is the Lewis variation. So good, we did get a variation, which is good. So nice. Yeah, he has like eight variations, that's true. Very true, 1,000 variations in this product. So does Verstappen, dude, Verstappen. I don't think I've seen every single Verstappen variation. This is unbelievable. Wait, so the Hamilton was or the Lewis? Hamilton. Well, they're both Lewis's Hamilton. No, yeah, Hamilton. Hamilton, there's a yellow, Joe Guan Yu, to 199 for Alfa Romeo. Kevin Bullock-Baschie at Max Verstappen. Lupe Drugevich, Carlos Sainz, Nicholas Latifi, Lewis Hamilton, Grand Prix winner. Yeah, I think it's okay. Yeah, I think it's fine. Colin Williams, Green, Carlos Sainz, nice one there for Ferrari to 75. Carlos Sainz, Green. Here, Gasly, Enzo, Fittipaldi. That is a negative, Sebastian Vettel, 96 out of 100. Actually, it's a sepia. That literally looks dead on to a negative. But nice one there, still a nice Vettel for sure. Lando Norris, Liam Lawson. That is a yellow, Mick Schumacher, to 199, for Haas. George Russell, and Carlos Sainz. Joe Guan Yu rookie, Joe Guan Yu rookie, Verstappen and Mike Crack. Ooh, orange, come on baby. That is Max Verstappen on the orange. Card number two, 16 of 25. There you go, congrats. Not quite the pink, but an orange will do, congrats. Zach, there you go, buddy. We'll get that one magged for you, my friend. Very nice, on the Verstappen orange. That will work. George Russell and Daniel Ricardo. Last pack. That is a big one for sure. Ricardo Drugevich, Sainz and Duhan. That'll do it for your break there, Zach. Thanks again, as always, for the support man on Personals and in Group Breaks. We'll get it right out to you, buddy. Congrats on that, orange, Max, pretty sweet.


Yeah, it's kind of like it's like kind of like the Pinocchio story like Pinocchio there's Gepetto the puppet maker the marionette maker and then and then Pinocchio it starts off as a puppet and then the whole thing is where Pinocchio wants to become a real boy and it's a drama you know he gets taken off into Pleasure Island and you know it's a nasty things happen just from wanting to be a real boy I had a friend of mine who after studying the course for like 15 years she would she was more and more aware that the body was like a puppet and her prayer to the Holy Spirit was get me back on the strings get me back on the strings it was the authority problem I don't want to be an autonomous person with an individual separate will and struggling to try to survive and make a living and you know try to fight off disease and death I get me back on the strings and oh great you know puppet master spirit you move the string you know like that old song you know if you want to see me do my thing pull my strings if you can sing that to the Holy Spirit you know get me back on the strings and you pull the strings and and let me live a life you know like St. Francis said make me an instrument you know let that be used that makes all the difference now the ego made up time and space it made up the puppets and it made up the concept of the doer you know in fact people have said why don't we call ourselves human beings we should call ourselves human doings that's a much greater description human beings we don't being what do we get a few moments of meditation here and alming and then we go oh there's the being but now let's get back to the important rat race and you know and doing doing doing do be do be do do do do do do shooby do be do you know it's it's everywhere even Sinatra just do do do you know he Sinatra one time he was he was singing I did it my way which is the most autonomous song ever and and he collapsed on the stage just went down with doing that when I heard that I went yeah isn't that typical I did it my way you know yes there were times I'm sure you knew when I bit off more than I could chew but through it all when there was doubt I ate it up and spit it out you know oh come on it's like boom ego had to go down he had to go down on the stage you know it's just not the way you know if there's no there's no surrender in it there's no humility in it there's no humbleness there's no put me back on the strings it's more like I am an autonomous being and I'm gonna do it my way I'm gonna die you know and I'm gonna die doing it my way and that's that's what the human condition it just enacts that now I need to do nothing is more is coming into that state of stillness where it's not so much that the body does something or doesn't do something but the identification with the doer dissolves away you merge what Sadari was talking about that simultaneous merging with everyone and everything and it's glorious because there is no sense of a doer there's no sense of individual responsibility what is our responsibility we're to be happy our creator created us to be happy and when we try to take on responsibility and duties and obligations that are based on this false self-concept of being a doer that of course we feel guilt we feel shame we feel inadequacy we feel like it's never enough we don't feel worthy of love we feel very unworthy we feel shameful because we've misidentified with the doer and we've forgotten about the beer the being so this is about being done through you have to go through a phase where you're done through where you're back on the strings for the spirits you know laughing through you and smiling through you and hugging through you and singing through you and dancing through you ah that's being done through in a way that just every time that it happens you become more aligned more aligned with with the the creator with with the source with coming back to that pure sense of being this so you know we we have believed that we are personally responsibility for our survival you know and the spirits like saying well you can change your mind about that you know you need to be nearer neither careful carefree you can cast your cares upon him for he cares for you you can open up to that divine inspiration and take away the the burden of being a human being trying to struggle and survive and make it I for example I always thought you know like when when we were in preschool they they taught us in kindergarten they taught us to share that's that's a spiritual value sharing you know we're here sharing this evening together you know June has opened up her home to share her home with us and her dogs with us and and then John brought food and sky showed up with some pizzas and that there's a spirit of sharing when I would do these talks and gatherings the only reason we have lights or that we record these things is for one reason to share it you know to put it out and to share it and then I've had a few people already here tonight they say oh I just love I watch watch your YouTube every day and I enjoy it and and sometimes I've heard people well I yeah before I go to sleep at night I just put on some of the mp3s and I listen my husband's like come on get that things off your ear but no I'm now know what I mean what world are you in I'm in the world I want to be in you know it's like you know it's like they just it's like giving your mind over and aligning with these ideas and feeling the resonance in your heart with these ideas and then and strengthening them in awareness and then practicing them in your daily life and the more you give them away the more they become your own you know they become truly your your spiritual reality so that's what's been fun for me is just trusting that everything that I need would be provided I didn't have some kind of a master plan for traveling the world I in fact talk about traveling the world on a shoestring budget that I didn't have CDs money markets things to invest in and this and this to go out and kind of do the hotel motel route it was more just going and okay place opens up here or somebody takes me in there or this or that just traveling more like Jesus and the apostles and then that just has continued and even you know people will say well you can do that you know in your local neighborhood or whatever but you can't do it overseas but then that started happening donated frequent flyer miles okay spirit apparently it's just props you know it's like we've all been raised with the mentality is you can't do anything in this world without having the resources ahead of time that they would call somebody a fool who started to go out to try to do something or travel or do something without having the resources already but it's like they show up they materialize as you go along if you're being guided by the spirit the spirit like no problem you need some props here's this it's only ego pride that says oh no I can't couldn't accept that couldn't accept that that's what I was doing at the early years oh thank you but I couldn't I am my own provider and no I couldn't all pride pride pride pride and then the spirits like would you stop that I you asked me to help you you know like the old story of you know the the one that guys in the boat and they said it was I had to face the same thing you know and no no no I couldn't no absolutely no no and it's just like just stop it get over yourself you know just would you accept help if you're given sharing and shining happiness and joy that's an important function so don't cut yourself short you know if there are things involved with that that would be helpful then then do that so that's why I've had fun doing all these talks and gatherings but I it's even more fun when you get back and you've got the recordings a little digital video tapes or the in the early days camcorders and recorders and then you put it up on the web and you share it it's fun to share it you know people do write emails back thank you thank you for sharing that it really touched my heart and you know it's like oh I it touched my heart it touched my heart to give it and extend it and you don't you know people have said David you just don't get how marketing works you know what you should do is you know stop doing things on donations and put out of start getting some products I remember the first time I went to Australia my friend Raj down there I stated is how he said of the gathering was great this was great this and this but where's your product and I said product what do you mean where's your product said you can't keep this up and go around without a product you've got to have a product you know you got to have something to sell and I was like no I don't really feel like I mean there's this joy in the moment like I've done it for just like 15 20 years where I went around and I didn't have product for 20 years it was fun it's working well you know you really should have a product you know or you know you put some but don't put too much free stuff on the web because if you put all free stuff then you can't eat and I said what do you mean I've been eating all along if I am starting to lose my appetite but but I'm getting into the divine mana but it's like but no I don't I don't really believe that that's the way that it has to go I think if if everything you know freely you have received now freely give if that was what the way show are taught us then why not try that out and you know all that I give is given to myself the course says you know if you keep getting into the joy of giving wouldn't you be blessed by receiving the gift of giving and I'm not talking so much material things because the joy is in the attitude you know that's priceless you know if somebody came up and said I can give you an enlightened mind for 1995 or something like that and it really worked you know it would be a good investment borrow the twenty dollars you know would do something you know to get it but but actually it's not that we can purchase it or that we can bottle it or we can take a pill for it or anything like that it's that we have to actually exercise it give it away you know just in the friendliness just in the good old wholesome honest friendliness we just keep giving it away and it feels good and we do more of it and even when the naysayers come and say yeah but you're not you know this is not going to sustain it's just going to be a flash and it's going to be gone and those people say no in this world it doesn't work maybe even had an economics I was in university for 10 years and I remember this man CA Helmolde the economics professor and he would write on the blackboard there is no free lunch and my whole life I've had thousands of free lunches just going and being happy or the you know you shine your happiness and people roll out their love and hospitality everyone wants to have a happy guest for lunch who doesn't want to have a happy lunch you know all of us would love to have a happy lunch so you know it just disproves there is no free lunch it's like when you're happy they're all they all feel free like free lunches and it just turns the whole thing around from from what it was before


Welcome back. We are at lecture 33 today. 33 is a good number. Larry Bird's number, if you came to my home and you went up in the room above the garage, the bonus room, you would see a life-size cardboard cutout of Larry Bird, six foot nine when his playing day. So that's how eat up I am with Larry Bird. We happened to go to, he went to Indiana State. I went to Indiana State.  All that from lecture 33. So this is just going to be an exciting day. We are in the middle of 7.8 in the supplement to the text. Second order, linear differential equations. These are non, homogeneous because we have something other than zero on the right side. So this was our last example yesterday. So we are going to pick up with this one. If we have trig, namely not all trig, let's just open it up to sines and cosines. If you have a sine or a cosine or sines and cosines, your particular solution looks something like this. A, sine of x, b, cosine of x, the reason being you are going to take the original function, so you certainly want sines to be present. You are going to take the derivative, thus you want cosines to also be present. They can generate sines and cosines, and we have second derivatives present. So we really want both these functions in the particular solution. We want to generate sines. Ultimately, we don't necessarily care about cosines. We want them to disappear. We can kind of force them to disappear by forcing their coefficient when all is said and done to be zero. So we want there to be one sine of x when all is said and done, having started with a sine of x and b cosine of x as being added together. I didn't mention the characteristic equation or the solution to the homogeneous part. Any questions about that before we kind of pick up from where we left off? Any questions? A couple of you came in late. Also, we will probably have time for a web assign question or two today. Certainly if it relates to this, I'd rather use your question as our problem rather than the problem that I have written out for us to go through. Issues, questions with that? Everybody okay with why we need this? Why we need to start with sines and cosines even though we only want to generate sines? So what's our next step? This is where we left off. Okay, take y prime of the particular solution, y double prime of the particular solution, and plug them back in. So we can equate coefficients left side and right side. So I'm going to rewrite the particular solution. You need its derivative. Derivative of a sine of x. Derivative of b cosine x. And if we had something other than just x here, we'd have to use the chain rule, so be prepared for that on other examples. Okay, derivative of what we just found. So derivative of a cosine x is negative a sine x. Everybody okay with those? What do we do with those? If I sound kind of helpless today, I'm just kind of trying to prompt some more questions. Some of you I think were drifting on me yesterday. And I need to kind of keep you a little closer. Okay, so let's plug this in for y double prime. Looks like we've got minus 10 y primes. We'll plug that in and plus 41 of these. Okay, so y double prime. Now we've got minus 10 y primes. We've got plus 41 of the original y things. And when we do all that we're supposed to somehow generate one sine of x, or just sine of x. What would be the next step? Distribute what after that? Group together like terms, right? Let's see how many sine of x's we have on the left side. Compare it to how many we have on the right side, how many cosine of x's we have on the left side. Compare it to how many cosine of x's we have on the right side. By the way, how many cosine of x's do we have on the right side? We don't have any cosine of x's, so we have zero of them. So we are going to have some on the left side. Alright, so let's say, let's see if we can do all this together. We want all of the sine of x terms. Looks like we have negative a of them right here. Looks like we have, what, 10 b of them right here. Is that correct? Positive 10 b and what? Plus 41 a. Plus 41 a. How many cosine of x's we have on the left side? What do we have? It looks like there's a negative b. It looks like there is what? Negative 10 a and plus 41 b. So what two equations result from distributing, gathering up like terms? We need to find out a and b. Those were the two unknowns from our particular solution. So what are the equations? Somewhat simplified. Negative, nice. We can put these together, right? So how many b do we have there? 40 b. That's got to be zero, because we don't have any cosines on the right side. And this is how many cosines we have on the left side. Other equation. That one by itself is not going to get us anywhere, right? 40 a plus 10 b equals sine of x. Equals one. So we have two equations and two variables. We don't necessarily love the coefficients that we have, but it's a doable system, small system of equations at this point in time in the problem. I don't know. What do you want to do to get rid of a's or b's or how do you want to approach it from this point? Substitution. I don't care. Lots of different techniques you choose. You multiply the first one by four. First one by four. So 160 b minus 40 a equals zero. And leave the second one alone, just rearrange it. So we want a 40 a, which we already have. We want it added to 10 b, so we didn't do anything to that equation. So if that's the case, if we add them up, we're going to lose the a's and we're going to have b's. So 170 b, these drop out, equals one. Be prepared for some mighty, ugly coefficients. We're going to have a couple on this problem. So it looks like b is one over 170. I was kind of thinking that when we started this problem. I kind of thought that the coefficient was going to be one over 170. Not really. You have no earthly idea what kind of hideous coefficients are going to get. What now? We have b, so we have one of the two values we're looking for. Go back into any of these equations. It doesn't matter which one you think will be the easiest. Let's plug into this one. So 40 a plus 10 b equals one. Does that look right? 16 17ths, which is four into there, well, eight into there, twice, and eight into there, five. Two over, what, 17 times five. 85. Another delightful coefficient. So we go back to the particular solution, which was a sine x plus b cosine x. We now know a and b. A was one over 170. Where is that? Yes. Now that's b. A was two over 85. B was one over 170. So our final solution, which we haven't messed with the homogeneous part today, but we got that yesterday, the homogeneous part. So we've got cosine of four x's and sine of four x's. Those are certainly different from sine of x's and cosine of x's. We add to that the particular solution. And if you have some leisure time today, take this equation, take its first derivative, and take its second derivative, and plug them into the original equation. And you'll see that all kinds of stuff on the left side drops out. And finally, when you've distilled it all on the left side, what are you left with? You're left with sine of x, which is what the right side was in our original homogeneous equation. So when in doubt, and if you have the time, you can check it. It does take some time and effort to check it, but it should work out to be the case. Any question there? I'm not remembering two over 85. Is that arithmetic correct? Anybody see an arithmetic problem there? Maybe I'm just not remembering that solution. I do remember the one over 170, two over 85. Anybody see an arithmetic mistake? I mean, I didn't I plugged it into the other equation. Okay. I didn't remember that answer. Questions on this process. So we have now looked at non-homogeneous second order differential equations that had exponentials in them. That was our first example. We did a second example that had an exponential, but we had to modify the solution to get one to finally work. This is our first example of trig. It's not a super complicated one, but it's our first example. We need to look at one that has polynomials on the right side. So let's do that next. Remember we're going to have this second order equation, some variety of y double prime, y prime and y on the left side. We're going to have one of three things. We're going to have exponentials, sine or cosine. We're going to have a polynomial. So it's not completely wide open on the right side. If you see a problem at the end of this section, and there are some that have, well maybe I was looking at another problem set. I think actually these are probably all fairly safe. But if you see one that come across one that has secance or tangents or some other type of function that we haven't dealt with here, then that requires a different method than we're covering here. This is kind of a brief survey. You'll get a more in depth study of these in math 341 if that's a course that you need for your degree. How many of you will eventually need 341 that you know of a full semester course in differential equations? How many of you think that you need two semesters? You also need 401? You need 401? A couple of you? So you're going to get a further study of this stuff in courses beyond this course. Alright, let's take a look at this example. And on the right we have a fairly simple polynomial. But it is second degree. And in fact that's what you have to think of when you see the right side. It's a second degree polynomial. What does the particular solution need to be in order to generate a second degree polynomial? It needs to be a second degree polynomial, but not just x squared. So let's get the homogeneous part out of the way first because it does comprise part of the solution. Characteristic equation? I'm listening to you. Doesn't matter. I mean r or x, we've used r because we've got our solution in terms of x so we wanted the r. Factors of negative 2 that will give us plus 1 in the middle. So the homogeneous part of the solution, two distinct real roots, even if you're trying to not, you're purposefully trying not to memorize this. By the time we get through this you will have looked at so many examples I don't think you can help. But correlate. What do you do when you have two distinct real roots then roll right to the solution? C2E to the negative 2. Everybody okay with that? Hopefully it will be second nature when test time comes around. A particular solution. We've got to generate some x squared term, some polynomial term of degree 2. So we have to generalize every type of polynomial that we put in. If, for example, this is a number, let's say that we've got a number over there and it's 5. Well, we generalize that to any number, which we don't know what number to start with. We would start with some number a. If that is, let's say, linear, we want to start with a linear expression, but we don't necessarily know how many things to begin with. So a generic linear expression would be ax plus b. So that's a kind of an open-ended linear expression. Well, we've got an x squared on the right side, so we need to generalize that to kind of any quadratic polynomial. So there's our particular solution. So we want to generate a quadratic, we better start with a quadratic. We don't know anything about the quadratic that we need to start with other than it needs to be some generic quadratic. So there's a generic quadratic. Now, before we go any further, certainly since y has an x squared in it and this has a y in it, that gives us the x squared term. So this term can generate an x squared term pretty obviously because it's in here negative two times. But in the other terms, we're going to have to take the derivative. Well, when we take the derivative of this, let's go ahead and do that, which is what? The derivative of this generic quadratic polynomial. Plus b. Is that all right? So we're going to generate some of those. Here they are. Well, we don't have any x squareds here. So what probably is going to have to happen to the x's that we have here and the x's that we have here and the constant that we have here and the constant that we have here? They're going to have to drop out somehow. Because all we want on the left side when all is said and done, the only thing we actually want is the x squared term. So it must be true that the x's are going to all knock each other out and the constants are all going to knock each other out. In fact, we have more stuff here. What's the second derivative? Two a. So we've got a constant of c, b and two a. We also have some other coefficients in here like a minus two y. Eventually we want all the constants to drop out. So we'll do what we normally do at this point in time. We plug into the original equation. y double prime is two a. We add to that y prime, which is two a x plus b. We add to that we subtract two y's and the right side we're supposed to, after we do all this stuff, we're supposed to generate an x squared. What do you think needs to happen at this point on the left side? Well, we can distribute the minus two, but after that, combine all like terms. What kind of terms do we have? We have x squared terms. We have x terms. And anything that doesn't have an x squared or an x is a constant, so we'll put all those together. So how many x squared do we have when all is said and done on the left side? That looks like a nice equation. What's negative two a have to equal? One, right? How many x squared do we have on the left side? Negative two a. How many x squared do we have on the right side? One. We have one x squared over here. So there's one of our equations. So in fact, we get a value out of that right away. One equation, one variable. How many x's do we have? There's two a x's and there's minus two b x's. Is that correct? And then everything else that doesn't have an x squared and doesn't have an x is a constant, so what are the constants? Two a plus b minus two c. So our three equations that result coefficient of x squared over here better equal the coefficient of x squared over here, which is one. So we get a solution out of that. A is negative one half. The coefficient of x over here better equal the coefficient of x on the right side, which is zero. We don't have any x on the right side. Let's go ahead and get a solution out of this, since we already know what A is. So we get a coefficient of x squared over here, which is one. Two a would be negative one. If A is negative a half, stay with me now. I'm starting to drift, getting that drift thing going on. So we have two negative two b equals one. So two b equals negative one b is negative a half. Does that look good? And two a plus b, which we know both of those now, minus two c. That is the constant on the left side. What's the constant on the right side? Zero. So two a that'd be negative one plus b, negative a half minus two c equals zero. So we get what? Four c three halves, negative three fourths. I was thinking that. I was thinking we were going to have negative three fourths for the constant. Right before we started this problem, and lo and behold that happened. I had no earthly idea. But I did remember that all three of these terms actually did appear in the particular solution. We really want a final answer, but let's go ahead and write down the particular quadratic, which we started out with an a x squared plus b x plus c. Now we know a b and c. So negative a half x squared minus a half x minus three fourths. So our final solution, what was our homogeneous solution? C one e to the x, C two e to the negative two x, kind of a lengthy solution. But I don't think getting there is like ridiculously complicated, it's just lengthy. It takes a while to get to these solutions. Question about any part of that? Easy, fairly straightforward. So now we've handled kind of an individual exponential individual trig function, or sine or cosine function, and now a kind of a generic polynomial. We can have combinations of those, which I guess that's probably where we need to go next. I wasn't going to do that next, but I think that's probably the appropriate thing to do. This is quite a combination of stuff on the right side. But everything is polynomial, exponential, or sine or cosine, or some combination of those. In this case we have a product of a polynomial, pretty simple polynomial, and an exponential. I don't know if we've had one like this. This kind of might be all kinds of stuff that we haven't encountered yet. Usually we have the y double prime, the y prime, and the y. So we're missing the y prime. That's going to change the characteristic equation, but it's still doable. In fact, what is the characteristic? Minus four. Minus four? Because we're missing the linear term, the first derivative term is missing, which normally translates to the r term. So we've got the r squared term, and we've got the constant difference of two squares. So we get two distinct real roots. Maybe I'm confusing all these answers today, but some of these answers are different than what I was expecting. I guess we're okay. Does that work? Now, you can take this all at once from this point. It can get a little messy. So I would recommend that you take the particular solution in pieces. The only way, these are so different. This piece of the non-homogeneous solution and this piece of the non-homogeneous solution, there's going to be no overlap. The terms that are going to generate the cosine of two x are completely different from the terms that are going to generate x e to the x. So you can take them in pieces when they're completely separate as far as what types of functions they are. So we'll actually do two of these and this combine our answer at the end, because that's all we do is add all our stuff up anyway. We add up the stuff that's going to generate zero. There it is. There's our homogeneous solution. We can add in our solution that's going to generate cosine two x and then we can add in our solution that's going to generate x e to the x. So all of it becomes part of our final answer. So we're going to do this in pieces. So let's do this one first. We've got to figure out how to get some x e to the x's. Well, we need a generic version of each one of those. This is a linear polynomial, first degree polynomial. So if we generalize that, what's a generic linear polynomial look like? A x plus b. A x plus b. And then we need a generic amount of e to the x's. And by the way, our e to the x's already part of the homogeneous solution. They are not, so we're okay to kind of go off with just e to the x's, since they're not a part of the homogeneous solution. So normally we would say we don't know how many of these to start with. Let's start with some unknown amount of these e to the x's. If this were by itself, that's what we would do. But do we really need that c there? Couldn't we distribute the c into here and get c times a? Well, we don't know what c nor a is, so isn't c times a some number? And c times b is again, we don't know what either one of them are, but c times b is some number. So we can really do without that. It's kind of already accounted for in the fact that we have unknown coefficient of x and an unknown constant. You can keep it if you want to, but it's just an added variable and it causes more trouble than it's worth. So our particular solution appears to be ax plus b, generic, linear, polynomial times some unknown amount of these e to the x's, which we've absorbed that in the preexisting unknown coefficients. Now, granted we haven't dealt with this yet, we're going to do that separately. So I'm going to rewrite the particular solution. So we really don't need the first derivative as far as subbing it into the equation, but we need to go through the first derivative so we can get the second derivative. So the first derivative, let's use the product rule first times derivative of second plus second times derivative of first. What's derivative of first? derivative of that. We've got the same thing again, right? So we use the same product rule first times derivative of second plus second times derivative of first. And then the derivative of this last piece, a, is a constant. We can bring it along or coefficient, bring it along for the ride derivative of e to the x is e to the x. So we need y double prime and we need y to plug back into the equation. Let's see if we can put some things together. I guess we can do most of that later. Certainly we've got a e to the x and we're going to add in another a e to the x. So we've got 2 a e to the x. So there is y double prime. We're going to subtract four y's and then we're going to attempt to generate this x e to the x term. So why don't we on the left side, we're going to have an x e to the x term. Let's kind of gather along how many of those we have. So how many of those do we have? So if we distribute this e to the x, so we've got a of them here minus 4 a of them here. I like the sound effects. Keep them coming. So how many of those do we have? How many x e to the x on the left side? We've got a of them here and minus 4 a of them here. Is that correct? Minus 3 a. What other terms are we going to have on the right side? Just e to the x terms. So let's gather those up. How many e to the x do we have? There's b and there's 2 a and here's a minus 4 b. So we had a b and now we have minus 4 b. So what? Negative 3 b? 2 a. And every term had an e to the x in it at least, right? So I think those are the only terms we're going to generate. X e to the x and just some number, arbitrary number times e to the x. I think we've taken care of everything. Does that look right? To you on the left side, x e to the x and e to the x are the only terms we have. On the right, we have 1 x e to the x. So what's an equation that results from that? Negative 3 a is equal to 1. 1? So a is negative 1 third and negative 3 b plus 2 a which we now know what a is. Better be what? Better be 0 because we don't have any of those over here. So negative 3 b plus 2 a's equals 0. So b is what? Negative 2 9ths. Does that sound right? So those were the coefficients in our generic linear expression that modified or was multiplied by e to the x. So at this point, we have our first particular solution. You know, I don't know what to call it. Let's call it p1 is a x plus b. So that is negative 1 third x minus 2 9ths times e to the x. So apparently if we plug that in for y and take its second derivative and plug it in for y double prime, we could generate this term right here. 1 x e to the x. Now we move on to the second piece of the non-homogeneous solution. So we did this already. So I'm just going to put a line through that. We've already done that but we haven't. What was it? Cosine 2 x. So we need a second part to the particular solution. Distinctly different so there's no overlap from the first piece. How are we going to generate some cosine of 2 x's? What could we call our original y thing in hopes that by throwing in the original y thing, actually negative 4 of them, and one of the second derivatives, we could generate cosine of 2 x. What's kind of a generic version that we need to use to generate sines or cosines or sines and cosines. Does that work? Now if it bothers you that we're using A and B again, it's a new one. So they're not at all related to our original A and B.  You'd have an A x plus B times e to the x plus c cosine 2 x plus d sine 2 x. So I think it's better to do them in pieces. You can do them all at once if you want to. Derivative of A cosine 2 x. Is that work? Derivative of cosines so it's going to be negative. Derivative of B sine 2 x. We really don't need that one but we do need the second derivative so we have to go through the first derivative to get it. Derivative of negative 2 A sine 2 x. Are there lots of opportunities for sine errors, numerical errors, and all kinds of stuff? There are along the way so it's not one where you can kind of lightly concentrate. Some kinds of problems that we do you can like. The homogeneous part of these problems. You could probably do them and you know ride the stationary bike at the same time or work on the treadmill. But this is probably not a treadmill problem because you're going to have some errors if you do that. Negative 4 A cosine what's derivative of 2 B cosine 2 x. Minus 4 B sine 2. Now if you want to kind of check down the line aren't you going to gather an extra coefficient of 2 along the way right? So you better have a 2 in each of these and you better have a 2 times 2 in each of these.  And every time you differentiate a sine you keep the same SIGN. So you can kind of check and it doesn't take much to check as you go down the page. Alright what do we need? We need y double prime minus 4 of the original y things. And what are we hoping to generate with this stuff on the left side? Cosine 2 x. We've already generated the other term on the right side. And in fact how many cosine of 2 x we've got one of them. So let's gather up the terms that we have. We have cosine of 2 x terms and we have sine of 2 x terms. How many cosine of 2 x do we have? Negative 4 A minus 4 more A negative 8 A is that right? And how about sine of 2 x terms? There's negative 4 B and negative 4 more B. Does that work? Negative 8 B. So what are our equations from this point? Negative 8 A better be 1 and negative 8 B is 0. So in order for that to be true B has to be 0. So we allowed for a certain term to be here. In fact it's not here. And we'll see if that kind of makes sense based on the left side of this equation. So our second piece to the particular solution what was A? A was the coefficient of cosine of 2 x and B was the coefficient of sine of 2 x. We don't have any of those so I'm not going to write it down. So there's our second one. Does it surprise you that we lost the sine of 2 x term? I want somebody to explain why that should not be a surprise. There isn't any sign on the other side. But what do we have on the left side that kind of made sure that we weren't going to generate any signs? It's going to go back to the same function. If we start with a what? A cosine and we take its second derivative aren't we going to be back to cosines? So we shouldn't have to mess with signs in this one. We only want signs present when we've generated them elsewhere and we want to get rid of them. Well we didn't generate them elsewhere. There's a cosine. The second derivative is cosines. No signs are generated on the left side. So it shouldn't be a mystery that we don't have any in our particular solution. Alright let's gather up our solutions and then we'll gather up our books and our gear and we'll leave. So what is the homogeneous part of the solution? Boy I got a quick answer on that one. Use that statement more often. Use it about halfway through the class and then just say now we're not leaving. E to the... Alright first particular solution. Wait a minute. What did we have here? Negative one-third. X minus two-nights. That was our first particular solution. So we did have a product there. We had a generic linear polynomial times a generic exponential and then we also had a completely separate term and it's permissible to do them separately. Alright I wasn't going to tell the laycomers but that's just not fair and I'm not that kind of person. We are not meeting tomorrow. That would have been kind of mean wouldn't it? And I will see you on Friday. Yes. That's kind of the next phase. Yes.


What's up everybody, I'm the Mangus, you are awesome, and the time has come to make us a Talos build without having any idea what his stats or damage numbers are. We're going to base this off of what we do know, and then go back to the build once the game is released to see how it works. From what we know of Talos' kit is that he's going to be a bruiser tank that thrives in the thick of team fights. With that in mind, I'm going to try to make a build that takes advantage of Talos' passive and active abilities to wreak havoc while debuffing the enemy team with the option to peace out if things get a little too rough for our liking. I'm making this build with solo lane in mind. As most of you know, Ethereal's item system includes 6 item slots with extra slots for boots and consumables, so we'll actually be building 7 total items for the finished setu. First up, we're going to go with Reinforced Boots. This will give us 60 movement speed, 25 armor, and 100 health. This passive will boost our movement speed by 10% when we haven't taken damage for 10 seconds. Talos doesn't have much in the way of an engage outside of his ultimate, so the extra movement speed will let us get all up on us before people can properly react. Our first item is Parhillion's Havok, no idea if I'm saying that right, an axe that gives us 40 physical power, 10 physical pen, and 10% cooldown reduction. I want this early to give us a boost through our wave clear and some bully potential. While this is our first item, it's kind of what the entire build is centered around. Talos has some big physical damage AoE attacks, and the plate shredder passive on this item will let us debuff a ton of targets at once. Second item is Berserker's Shield. As I mentioned in the Iran build video, this seems like a must for any type of bruiser myth. It gives us 50 armor, 300 health, and 30 power. The Vanguard's Rage passive provides us with power and attack speed while above 50% health. Dipping below 50% converts that speed and power to armor and health regen. I think it's pretty self-explanatory as to why any melee bruiser would want this. The next item's build order kind of depends on your scavenging huntsman passive. I have it as number 3, but that's probably a bit optimistic. Harvey's Backplated Chestguard gives you 60 armor, 30 magic resist, and it converts 5% of your armor and magic resist into physical power. With scavenging huntsman passively providing talos with increased protections, this thing will probably be an auto-include in any type of talos build you do. Next slot is a bit of a flex slot I think most of the time I would take Kordering's Edge. This gives us 40 physical power, 24 physical pen, and the passive applies a debuff to enemy health and shield regen. This type of item is usually a must have in any MOBA. However, if no one is building lifesteal or something crazy like that, you may want to look into some of the other axes. Our fifth pick is Amulet of the Ancients. This will give us 25 magical power, all it's really going to do is increase the bleed damage a little bit from Radial Sever, but it also gives us 400 health and 20% crowd control reduction. The passive, called Ancient Salvation, activates when you dip below 30% health. You spread roots in a 150 unit area that root enemies and heals you for 2.5% of your max health for each enemy hit by the roots. If they hit enemy minions, it also heals you, but only for 0.5%. This is going to be our decision point indicator.  The final item is Cursed Gift. This gives us 25% mana regen, which we couldn't give a shitless about. 15% crowd control reduction, 20% cooldown reduction, and 10 movement speed. This item has several passives, with the third being the reason why we take this last. The Golden Brecquium passive gives you increased passive gold gain at the cost of halved economy from lane and jungle minion kills. We do not want this in the early game. What we do want are the other two passives in the late game. Eldritch Boom gives all allies around us 10 mana regen, 10 health regen, and 13 movement speed. Cursed Presence gives all enemies around us minus 15 armor, and minus 15 magic resist.  And that's my possible Talos build. Some of the weaknesses I see are that we don't have much magic resist, so this build may get blown up by mages until we can get our passive to max stacks. We also don't have much persistence, so we'll be susceptible to crowd control. Let me know what you guys think of the build, tell me what you would change, and how you would build Talos. Be sure to check out Jelly Knees' website, etherean.gg. He's been working hard on creating a myth builder that will make all of this so much easier. I don't know if it's functional yet, but I'm sure it's going to be a tool that you guys will love. That's all I have for now. Like the video if you want, sub if you feel like it, but for now this is the mangoes starting off. You guys, have a good one. Mango!


The question was how wide is your trouser leg opening? Well, let's find out. Okay, so here's a pair of trousers. I've got my tape measure and you can see that I am measuring seven and a quarter inches, which when doubled means that my leg opening is 14.5 inches, about 37 centimeters if you use the metric system. For reference, I am five foot nine inches tall, currently rolling in the low 160s for weight. And I mentioned that because it's very important that you dress your body type. Everything with tailored clothing needs to be in proper proportions to look good. So even if you like the way my trousers look on me, if you're not the same size and shape as me, the measurement is absolutely meaningless. So again, know your body type and dress your body type to look and feel your absolute best.


Hi guys, welcome back to this channel and in this video we are going to talk about FIFAL World Cup Tickets.  Remember I told you Aurora is up to date and what we have right now is the last phase ticket is also announced. Meaning that starting 27th of September they will have what you call the last phase sale of the FIFA World Cup Tickets. Remember we are talking about these FIFA World Cup Tickets you need to apply online you want what you call the FIFA World Cup photo and that is what you call the fifa.com dot tickets. That's why you need to apply for the favorite tickets and something that still I need to let you know is that these tickets we are talking about they are divided into what you call category fours that divided into four categories. We have category one, we have category two, we have category three and also we have category four. Depending on how your pockets stand at that time and how much you are waiting to buy it for that ticket but these tickets are priced accordingly to the different categories and you find that most of these tickets that are on last sale ticket we are looking at tickets that are going to be high priced. For example we look at tickets in category one they will be a little bit high priced because itself you are looking at category one whereby you are going to be sitting or you are going to be sitting yourself in what you call the prime areas the VIP areas at one world another and the latest announcement we have that starting midi day on 27th of September the sale will be out at that time and remember it is the same time that we say for residents or for people living in Katal if your residents with a valid Katali in Katal you will only you only reserved for what you call category four so you can buy a ticket for yourself in category four which is not going to be very expensive depending on actually even also it will definitely depend on what kind of match you want to watch at the same time it's also another issue that you also need to consider but according to the supreme committee for delivery legacy that is the four world cup they are saying category four of the four world cup last minute sales tickets is only reserved to what you call the residents or the people living in Katal with a valid Katali. Hope I'm trying to share something for you and your hope you definitely now see that you have it and something that still need to know that people who are coming to watch matches for the four world cup using what you call the higher card if you are coming from outside the validity of the higher card will be on 23rd of January meaning short of that you will have from 1st November until 23rd of January to be in the country that is the duration that your higher card can hope you apart from that you will have to exit the country and if you don't exit the country then you will go what you call legal measures will get against you that is what I was telling people who are probably saying that after the fifth world cup I'll take to hide out myself yes you definitely try to hide out to yourself but when they have you and they take away it will not be good for you it will be trouble for you in one month which is quite something that really understand them. Hope I'm trying to share something for you again by the way feel free to do by commenting in the comment section I'll always be more than happy to get back to you remember it's mixed from the next creation TV