Welcome back it is Thursday and that means acting analysis for animators and today I'm going to take a look at episode 2 of Killing Eve Can't wait to watch season 2. It's not out yet for me. I can't watch it yet But right now let's dive into episode 2 of season 1. Let's check it out First up is this walk and this is just an example for contrast where you have a character that knows what she's doing She's very confident while she has her arms down It's very tense and very nervous and you can see how she's barely moving the head You can see the rhythm how there's a lot more up and down She has a swing to her arms much more relaxed compared to her who is stiff She looks down. It's a stiff head. She doesn't quite know what is going on You can see this in from behind or you have those quick little steps Again, she doesn't know where she is what she's doing She looks around and this is all serving the contrast often You can see this here if I go back look at her shoulders How they go back and forth in Y if you're an animator here in Y Left right left right very relaxed same thing here here You can see even though the arm is slightly cut off even though, you know I'm a fan of gestures and movements that are not completely in frame, but you can see that relaxed Walk, you know the steps taking out the keys. It's all very relaxed Whereas if you go back to her on this side here looking around quick steps Much more contained you can see this here too in terms of contrast as she trails And then has to kind of catch up and all those little fast little moves again Just contrasting your characters if you do have two characters think about how could you contrast them in terms of posing in terms of Movements how fast they move just tell us something about how they feel so we know their difference Unless of course the similarity of a character is you know the the main point of the shot Of course, there are always exceptions to all my examples But if you have multiple characters try to make them distinct and different if that's what the shot calls for But think of it in terms of posing and movement and gestures and just how fast which is everything that they do is Based on how they feel their emotional state what happened before to them or what they think is going to happen afterwards Their goals their actions all of that goes into your posing and your movement in terms of timing Same example is this one where they talk and he wants to tell her something that might not tell her something That is not good news to her so you can see that reaction here and she doesn't really want to hear it So he starts off with what he wants to say and turn the sound back on so you can hear The interruption as he starts going and then you can see what she does This is so good I turn the sound off so what about this is that you can have lip sync like between Her and him they can both talk you couldn't just be just one character But it could be a lip sync that's just for one character and the character tells someone what they want of them which is exactly this example and This person doesn't want to hear it now you can do this with no interruption So this is the length of your shot no interruption in your lip sync or what you do is you have your character do something I'm also a fan of characters doing secondary actions right where you can have your lip sync in your dialogue and interaction everything But the character is doing something while performing those lines So now she can use this since that's the setup of the scene there. She can use this to do this So she interrupts the whole thing you can put in an artificial pause in your lip sync and in your dialogue and in the Whole shot and now what I love is that that look and this that it like yeah Yeah, I did this I don't want to hear this and because you are artificially adding this and adding a break in the lip sync Now you have the option to do whatever you want acting wise It's your creative freedom there and you can have that a look there You can even go back to just a staring contest of that and I love this and I love that a look here Just when we hear that the blender off-screen is done. She has a little look up Okay, I guess I'm done now and it's so subtle or you have even a little bit In the eyebrows launches again, just watch your eyes little face It's very subtle and I love that quick little look as if she's happy again I'm done and then it continues with the line But it's such a great moment of using secondary action using props to kind of change the shot if needed I mean in this case obviously it's part of the scene But again if you have a scene where it's there's audio and you want to break it up You want to kind of change this and not make it you know the kind of the same as the the source Then maybe think about where could you place your characters? What's environment? What are the props? What can you do to artificially add potentially a break and where the characters have to stop now? Of course, this is tricky if someone talks in the line that you suddenly pause and then if it feels like you're hitting pause on Audio so you have to kind of find a way maybe maybe kind of edit the audio to make that work I'm just saying this could be if it's possible for you to do that an example of Taking an audio piece and then really making it your own changing it changing the subtext Using the environments and using props that could really make it your own in your own more creative and original Interpretation of that audio piece next up is something else that's very subtle But I love every single moment that she does here So she puts on this mask but actually she concoct something that's not very healthy there And this could just be something where you put on the mask and you think about constraints You're thinking about you know handling this you probably have some spline controllers to do this And just probably concentrating on intersections and all of that stuff which is understandable and how this really folds around the head This would be very very tricky to do in animation. But what I love is what she does here This just that moment that even though she prepares something that's probably given a character very toxic very lethal She's preparing something where someone is going to die is very serious and she has a moment of that's kind of Kind of a pain to put that on just that moment of Levity of humor and of course you can do whatever you want to but even in something like this where it's very technical where you Interact with props. You can think about what could I do character wise that makes it funny sad Whatever you want to do But it gives us a little bit more of an insight into the character and almost distracts from this overly technical aspect of animation Where it's just really character driven and I love that just that a look the winding of eyes went to blink And then she gets back It's also just the contrast of that slow now. All right. Hmm. Okay. Well, I'm ready So good and staying within the the subtleties and breaking up of shots. Here's something Jesus cracks me up. Sorry. Oh, it cracks me up. So this is the line where she asked Can you do this for me as possible?  I can do this for you and it's kind of to impress her Or just somewhat braggy and she just she's just not buying this right. She has that look of really and you got that I rolled great besides that awesome that Just that look is great. This is something else where let's pretend This is your shop and you get two characters and the contrast of you know, the visuals and everything standing sitting But it's a longer piece and you don't really want to have one long piece as we go back You can see it's a long longer piece where they talk So what could you do to break this up? It's not so potentially boring or just kind of static if they say something that Warns a reaction. You could cut away to another character And then this character could be you know, if pretend that this audio piece is serious Then this moment could be the moment of comedy now again, you don't want to copy exactly the shot But you can think about do I have a moment in my line where they say something where I can cut to someone else Reacting to that line and this could be with an eye roll. This could be with a laugh This could be actually turning towards them because they're not suddenly interested This could be them talking about something very important or about money like a heist or something in a coffee place And this is a person drinking coffee over here in the conversation going what are they talking about? So you can kind of add interest with other characters that maybe Heighten the stakes, you know if they talk about a crime and this character is listening That makes it potentially more interesting and dangerous that they're talking about this in the open in the public with other people So think about that too if you have a longer shot Maybe there is a way to break it up and it doesn't have to be you know Five shots and make this a whole sequence but just a cutaway to character listening and reacting It gives you the opportunity to do a lip sync You know all that performance that's tied to the lip sync and then cut away to a character listening and reacting with pantomime And it's not part of the audio at all and again You're bringing your own creativity and your own acting into a piece where usually lip sync you are tied to the length to the inflections The rhythm and again You can just add your own things to it and potentially also do less lip sync if it's you know too much work Where you cut away for a couple seconds That's couple sex you don't have to do the lip sync and just the character listening and eye darting and emote and processing and looking around I think that could be really fun and again You can change the idea of the shot the intensity of the shot What do you want to do with or whoever you put into that cutaway into that second shot potentially, but there you go That's it for episode two I don't want to go on more because the show in general is awesome You should just watch the whole thing if you can but if you think that this was cool and interesting And you want to use that in your shots as always I have workshops you can sign up We can work together on implementing these kind of ideas into your shots to make your shots potentially even better So let me know email comment What do you want to do? Link in description for all information regarding my workshops if you do want to sign up Also, if you want to you can subscribe and hit that bell button because I do upload almost every day So if you want to get notified about all my uploads Feel free to subscribe if you want to if you're still watching after my old rambling ending Thank you very much. I appreciate it and I will see you in my next clip


She's having a hard time with it too. Lecture 32, we are going to start non-homogeneous equations today. So we're in 7.8 of the supplement. It's not like we can completely forget all the homogeneous stuff that we just battled through in 7.7 because those solutions to the homogeneous equation are part of the answer to the solution to the non-homogeneous equation. So it's got two parts, two pieces. It's got a homogeneous part, which is kind of the way we've been doing thus far in this supplement, as well as what's called a particular solution. So the key difference you're going to see is that we don't have zero on the right side of the equation. We'll have something else. It is a select subset for this course. Now, when you take differential equations, that will probably open up significantly. But we're going to look at exponential functions on the right side of the equation, polynomial functions, and not even all trig, but sines and cosines and trig. So we'll have a select subset on the right side. Before we leave this, we will revisit this. We came up with this yesterday. And there's one more thing I wanted to do with it at this point. And then we'll revisit it. We really want to graph sine plus cosine. So what I did is I graphed in green. I graphed the sine. And in red, graphed the cosine. And then we added them up graphically. We added the y values at each individual x value. So we added 0 and 1 and got 1. We added the square root of 2 to the, excuse me, square root of 2 over 2 to itself. And we got this value. So we added them up. The reason is to get a look at this function, it is an oscillatory function, just like a sine or cosine would be. In fact, let's do this before we leave it. We could come up with an equation for this in terms of sine or cosine. Let's, in this case, use the sine. What would the amplitude of this new function be? What is, here's its amplitude in a sense that it oscillates that far above its axis and that far below its axis. What are those distances? Square root of 2. So the amplitude is going to be the square root of 2. And let's just, we could call it a sine or a cosine and just adjust it. But it's a little easier to adjust it if we call it a sine from this picture. Whatever is the period of either one of these, or in fact, both of these, is also going to be the period of the function when we add them together. So we don't need any special coefficient in front of x because we add x here and x here. So we're going to have x down here. So it says the period is still 2 pi. Can you see what kind of phase shift this one would undergo? If we brought this back here, pi over 4 to the left. So how do we accomplish a shift of pi over 4 to the left? Plus pi over 4. So there's the equation in terms of a single trig function, in terms of a sine or a cosine, in this case a sine, that is exactly the same as sine plus cosine. This will be called a phase amplitude form and we'll revisit that later in this material. But I didn't want to leave that example without getting that sine plus cosine in terms of a single sine or cosine function. So that's not really 7.8, but it's something that will come across different points in time in this section, in this supplement. All right, 7.8, non-homogeneous linear equations. So we're going to have these things on the left side that look very familiar to what we've had to this point. We're going to have some y double primes, some y primes, and some y's with the possibility of different coefficients. And then we're going to have something now non-zero on the right side, so it's non-homogeneous. So there's the main culprit for why it is non-homogeneous. We'll have in this course, since this is a brief survey of second order differential equations, we're going to have a limited set that we're going to look at that will appear on the right side. We're going to have polynomials and possibly some products of these things, too. Polynomials, sine and or cosine. So it's not even all trig, just sines and cosines, and exponential functions, e to the negative x, e to the 4x, e to the 2x, some type of exponentials. The method that we will spend our time with in this section is called the method of undetermined coefficients. That comes first. In 7.8, there is a second method, the method of variation of parameters, which we will not do. So just the first of the two methods in this section. And it's sufficient to do any of the problems that we're going to need to handle from this particular subset. So how do we go about this? This kind of summarizes it, and then we'll look at some examples. As it is a battle, the homogeneous part will not be a battle, I hope, because we've done that. We've reviewed that. We'll go back to that. But here are the things from 7.8 that are the highlights, and then we'll go back and look at how this plays out in an example. So there's the kind of thing we're going to encounter. The general solution of the non-homogeneous differential equation can be written as. So here's what our solution is going to look like. This is our final solution. It's going to have two parts to it. I tried to change that to an h, y sub h. That's going to be our homogeneous part. So if the print is small, there's y sub h. So we'll get the characteristic equation. We'll find the solutions. Then we'll go to the homogeneous part. That will be part of the solution. And then we'll also generate this new part called the particular solution, y sub p. It's the solution that's actually going to generate the stuff that appears on the right side. This might be an unfair question, but hopefully it's not. What will the homogeneous solution generate on the right side? When we put it in for y and y prime and y double prime, what happens to everything on the left side? So what does it generate on the right side? No? Why is it called the homogeneous solution? Because it generates 0. So this doesn't generate anything on the right side. So if we have something other than 0, we're going to have to come up with something in addition to the homogeneous solution, because all it generates is 0. So the particular solution is going to generate the stuff that actually appears on the right side. But that's why the homogeneous solution is part of the answer. It just doesn't generate anything. Then the particular part is the stuff that actually generates this g of x that's on the right side. Y sub p is a particular solution of equation 1. And y sub c, we've called it y sub h. The homogeneous solution is the general solution of the complementary equation. We didn't call it that. We called it the characteristic equation. Or the auxiliary equation. It's got lots of names. All right, so what are we going to deal with in this course as a subset of these things that appear on the right side, these g of x things? c sub n, I think in your book it says c sub 1, x sub n. But the c's can be different. So let's call that c sub n, x to the n, and so on. c1, x to the 1, and down to c0. Well, that would be a polynomial. So notice the vagueness of this solution. First, try y sub p. Well, that kind of gets you a little insight into the fact that it may not work all the time. And we're going to have to adjust it from time to time. But that's a good start, is to try something that, to generate c sub n, it may not be the same number in the particular solution. So they call it a sub n. But to generate some x sub n's, we better start with some x sub n's. To generate some x's, we better have some x's. To generate a constant, we better have a constant. But it may not be the same constant that appears in the g of x here on the right side. So we possibly will have a polynomial there. Here's our exponentials. We're going to have so many e to the kx's. How do you generate e to the kx's? You better start with e to the kx's. We may not start with the same amount we're going to finish with, because we've got to do some stuff with them on the left side. But we probably will be smart to start with some e to the kx's if that's what we want to generate. This gets a little bizarre, but I think you'll see in our first example today that if you want to generate cosines, you could actually have cosines and or signs. Where's that come from? How could you generate cosines by starting with cosines and signs? What are we going to do with the stuff we start with? So here's our y sub p. What do we do with our particular solution? We take its second derivative and plug it in there. We take its first derivative and plug it in there. We take the original function and plug it in there. When you take first and second derivatives of cosines and signs, what do you get? Cosines and signs. Cosines and signs, right? So we're kind of trapped in that cosine and sign area. Is it possible that you could have some cosines and signs, and the signs would all knock each other out and you'd be left only with cosines? It's possible? Is it possible you could start with cosines and signs and you plug in here, second, first, original function, and you could be left with just signs? That's possible? Or is it possible you could start with cosines and signs, plug them into the left side, and you'll be left with cosines and signs? That's also possible. So anytime you see a sign or a cosine or a sine plus a cosine, this is the type of particular solution that you use. So we allow for both, even though only one might appear. And there's always a disclaimer. So here's the disclaimer. Modification, put a big star by that. All this is different places in the supplement. But here's a key thing to remember as far as modification. If any term of the particular solution, so we're trying to generate this list, our particular solution, is a solution of the, we called it, characteristic equation. Multiply the particular solution by x, or by x squared, or by x cubed, or by x to the fourth, as many times as we need to multiply by x until it no longer is a solution of the characteristic equation. If it's a solution of the characteristic equation, what will it generate on the right side? It'll generate zero. So we don't want that. We want to keep kind of upgrading it or changing it or modifying it till it generates something other than zero. So that doesn't happen that often, but enough that we need to pay attention to how we modify our solution if, in fact, it's part of the complementary equation, or characteristic equation. All right. Let's see if we can get an example of several things today, see how they go. These are a battle, and you have to kind of make sure your derivatives and second derivatives are correct, and you plug them in correctly, and you multiply them by the proper coefficients, and you group together like terms. There's lots of potential places for error, so have to be careful. A little more so than just the homogeneous equation. Here's a good one to start with. So you can see that it's non-homogeneous. We've got e to the 3x on the right side, and it's a pretty good first example because we've got 1 e to the 3x on the right side. So we'll do the solution in two parts because the final solution is the particular solution plus the homogeneous solution. So we'll do the homogeneous solution first. Treat this as if it were a homogeneous equation. So we're really going to do that. Because is it 0 present on the right side of the equation? It's there. It's just got something added to it. Isn't there a 0 here? e to the 3x plus 0. There's a 0 there. So this deserves to be part of the solution. So what would y look like if we wanted everything on the left side to kind of drop each other out and we'd be left with nothing? Well, we go to the characteristic equation, which is 4 and 2, I think is correct. Plus 4 minus 2. Does that work for the middle term? So one of the r values is negative 4. The other r value is 2. We have two distinct real roots. What is the homogeneous solution? What's y sub h? Does that look right? So we've got three cases, right? This was our so-called first case. We shortcutted a lot of things. We went directly to the characteristic equation. And then once we got the two distinct real roots, we went right away to the nature of the homogeneous solution. So if I were to check this, the problem's already long enough, so we're not going to take the time to check it at this point in time. But if we were to plug in this thing for y, take its first derivative and plug it in up here, and take its second derivative and plug it in up here, what would happen with all this massive stuff on the left side? It would all drop out. You'd lose all the e to the negative 4x terms. You'd lose all the e to the 2x terms. Everything would drop out. You'd get 0. So that takes care of this part of the right side of the equation. Now we actually want to generate the non-zero part. So we want to generate some e to the 3x's. This, by the way, will be part of the solution. It's this part. So here's the so-called new part for today. We want a particular solution. What would you have to start with if you plugged into an equation the original starting quantity plus its derivative plus its second derivative, and you want to generate some e to the 3x's, what would you have to start with? Some e to the 3x's. Aren't they their own first derivative in a sense and also their own second derivative? When you take the derivative of e to the 3x's, you get e to the 3x's. And when you take the second derivative of e to the 3x's, you get more e to the 3x's. We don't know how many to start with. We know eventually we want to generate 1 e to the 3x. That's how many e to the 3x's are on the right side. So we start with some arbitrary amount. Let's say a of them, and I overlook something. I went a little too quickly to this part of the solution. I guess now is a good enough time. What was the modification? When would we need to modify that question that I just asked you about if we want to generate e to the 3x's, what do we start with? We start with e to the 3x's unless what? It's part of the solution. Unless it's part of the homogeneous solution. Are e to the 3x's do I have any here or do I have any here? No, we're OK. We've got e to the negative 4x's. And e to the 2x's, we want to generate some e to the 3x's. That's a different animal from the 2 that we have thus far. So we're OK. This is probably going to work. To see what a is going to need to be, we're going to have to plug everything into the left side, see what a needs to be in order for us to generate 1 e to the 3x on the right side. So if we're going to try this as our particular solution, what is the first derivative of the particular solution? Good. And we also need to know the second derivative, 9a. So do you see that we've got some e to the 3x's going? That's the only way we're going to be able to generate some e to the 3x's is to start with e to the 3x's. All right, so let's plug these things into the left side of the equation. The second derivative of the particular solution plus 2 times, according to this, the first derivative minus 8 of the original y things. So now we're actually going to equate left side and right side. So we've got e to the 3x's. How many do we have on the left side? We've got 9a plus 6a minus 8a. 7a. So on the left side, we have 7a e to the 3x's. On the right side, we have 1 e to the 3x. So what's the equation that results? 7a better equal 1, right? If that's the case, we better have the same amount left side as we do right side. It's an equation. So if 7a is 1, then a is 1 7. How many of you were thinking that we needed to start with 1 7th e to the 3x in order to generate? No, you weren't. No one was thinking that. I wasn't thinking that. There's no way to know that.  We have 7a on the left side. We have 1 on the right side. Well, a better equal 1 7. So our particular solution, it was a e to the 3x. We now know what a is. So there's our particular solution. Right above it is the homogeneous solution. So now we go to our final solution, which is their sum. So y or y of x is kind of up to you. I'm going to write the homogeneous solution first. There's the homogeneous solution. And we'll add in the particular solution, which is that. So there's the solution to our second order linear non-homogeneous differential equation. If we were to plug all this in, it would generate 0. On the right side, if we were to plug this in, in fact, we just did that. That's kind of how we arrived at the 1 7th. If we were to plug that in, its second derivative in the left side, this would generate 1 e to the 3x. And that's generating everything we need to generate on the right side of the non-homogeneous equation. Any question about any part of that? Anything that we did right there that needs some clarification? That was not bad. Told you we'd start with a pretty tame one. This is going to look equally tame, but it will not be. So it looks like the same kind of deal, except that instead of 1 e to the 3x, we have 4 e to the 3x. Can you see why this is going to be worse? Anybody? OK. The characteristic equation is going to have a double root. And in fact, e to the 3x is going to be one of the terms of the homogeneous solution. Then we have to jump down to the bottom of the page and do that little modification of our answer for the particular part. All right, homogeneous part. So we're going to treat this as if it were homogeneous. Tell me what to write down next. It's our characteristic equation. Factors of 9 that will get us negative 6 in the middle. OK, same. So here's our other type. So now we've encountered two of the three types that we have as far as the homogeneous solution. So r1 and r2 are the same, so we have a double root. So the solution, the homogeneous part, what's it look like? e to the 3x plus c2x e to the 3x. That all right? Done with the homogeneous part. Unfortunately, that's not going to generate anything on the right side. That's going to generate 0. So now we've got to worry about generating 4 e to the 3x. So how do I generate? This is going to be wrong, by the way. Go ahead and preface the fact that what I'm going to write down next is wrong. How do I generate some e to the 3x's? Well, I'm going to start with some e to the 3x's. That's what we did on the last problem, and it worked fine. Not going to work fine on this problem. What's the problem? What's the difference between c1 e to the 3x and a e to the 3x? There isn't any difference. They're the same kind of animal. If we put them in, we're going to get 0. So that's not going to work. What are we supposed to do according to that modification? Again, we don't know how many to start with, but we are kind of told or asked to multiply by x if that's not going to work. What about a x e to the 3x? We've got some of those in the homogeneous solution as well, so that's not going to work. Do we have any of those in the homogeneous solution? Don't have any of those. Now, that's not guaranteed that it's going to work, by the way, but we know for a fact that the first two didn't have any chance of working. What would be our next step now that we've decided that we have at least a potential candidate here for the particular solution? Take first derivative. Ooh, that's bad. Well, it's OK to have some of your terms drop out, but what we don't want to happen is all of them to drop out. So I don't know that that's going to be the case, but you're probably looking ahead saying that some of them are going to drop out, which will be the case. Or is that the same as like a C2, and your second derivative is just going to have the constant? Yeah, that's what I was saying, because it's just going to. Well, now wait a minute. We've got a product rule here. So it's going to get considerably right. We're going to have some keepers. We're going to also generate some terms that we're going to lose along the way. So we've got a product rule. There's the first term of the product. There's the second. So the first times derivative of second plus the second times derivative of first. That look right? There's derivative of second. Second times derivative of first. 3a x squared e to the 3x plus 2a x e to the 3x. What's next? Second derivative. So the derivative of this. We could put those together and factor out an e to the 3x. Let's just, I don't know. Do you want to do that first? Factor out an e to the 3x in these two. Because we've got a product rule here and another product rule here that might, doesn't matter that much. Let's just take the derivative. There's our first product. First times derivative of second plus second times derivative of first, which would be what? Now we move on to the second one. There's the first term of the product. There's the second term of the product. First times derivative of second plus second times derivative of first. What's derivative of the first? 2a. OK. Every term has an e to the 3x. Is that correct? Let's factor that out. And we have a 9a x squared. Do we have any more x squared terms? So we have a 9x squared. We may want to undo that factoring out of the a, but it's OK right now to do that. 6ax and 6ax, is that correct? Added together this term and this term, 12ax. We factored out the a. 12x and then what? 2a factored out the a and the e to the 3x, so just 2. All right, so I told you these would get kind of complicated. We still don't have the particular solution. In fact, we don't know that this is going to work yet. So we need to take the second derivative, the first derivative, and the original particular solution for y, plug them into the original equation, do what it says to do on the left side, and then see if we can generate 4 e to the 3x. All right, so we need y double prime. I'm going to put the a back in there. Why would I want to put the a back in here? Well, there's a chance that other terms are going to have a in them as well, but don't we really care about how many e to the 3x there are? So we're equating the coefficients of e to the 3x on the left side with e to the 3x on the right side. So we kind of want them to have a's in them. So y double prime, e to the 3x. So there's y double prime. We have minus 6 y prime. So y prime, let's go back up here. If we factor out e to the 3x, what do we get? And the rest of this is plus 9y. So we go back to our original particular solution, a x squared, and we've got an e to the 3x right here. Now when we do all that, when we put our particular solution in, excuse me, put the second derivative of the particular solution in, minus 6 of the first derivative of the particular solution, plus 9 times the particular solution, we should generate what? 4 e to the 3x. Can we on the left side factor out e to the 3x out of every term? Is that correct? So if we do that, and let's try to group together like terms, let's look at all the a x squared terms. We've got nine of them here. We've got what? You've got minus 18 of them here, a x squared, and what? 9a of them over here. Is that right? So did we lose the a x squared? That's a bad sign. That's a real bad sign. But let's keep going. How about the x terms? We've got 12a x minus 12a x. Is that it? Another bad sign. And then what else? We've got 2a minus 12a. Oh, that's got an x. So is that the only one? That may actually be salvation for this problem. In fact, it's kind of what we wanted to happen. We don't want any x squared e to the 3x's, do we? We don't have any of those on the right side. We don't want any x e to the 3x's. We don't have any of those either. So we kind of wanted them to drop out. Although it's kind of, if they all drop out, then that solution didn't work at all. So we lose the a x squared e to the 3x terms. We lose the a x e to the 3x terms. Now our left side is really what? What's left on the left side? e to the 3x times what? 2a? That was our only non-x, non-x squared term. All that stuff. Right, but we just took them in groups. And we looked at the ones that had x squared in them. We had 9a minus 18a plus 9a. So they all dropped out. But what about like this minus 6e to the 3x? We distributed that 6, didn't we? Did we distribute the minus 6 to this guy and get minus 18 here? I think we took care of the minus 6. And the minus 6 times the 2a here gave us what? Minus 12a right there? So we took care of the 6. And if the e to the 3x is bothering you, we factored that out all the way out in front of everything. So we took e to the 3x completely out to the left because it was here and here. And I left it out, but I put it back in there. So that. So I think we accounted for everything. So I think this is the only thing that's left on the left side. And on the right side, we've got 4e to the 3x. Let me take an aside here. If we ended up with everything on the left side canceling out, which we were well on our way to that happening, what would we have to do? What if we lost everything on the left side? Do it all over with x to the third. Right. We'd have to go back to the particular solution. We started with a e to the 3x. Then we went to ax e to the 3x. Both of those were scratched. Then we said, OK. What about ax squared e to the 3x? It has a chance. But if everything dropped out, we would have had to go back to the drawing board and make a new particular solution. So you kind of continue to modify or upgrade till we get what we want on the left side. Yes. What happened with that mean year? I'm a genius. The solution was wrong? No. That's really a great question. It just means that because of the nature of what you were doing on the left side, how many second derivatives you had, how many first derivatives, and how many of the original functions, what we thought would be a good solution that was safe. In fact, it was also not good. But it doesn't necessarily mean that it's part of the homogeneous solution that we generated. Technically, it is part of the homogeneous solution, but not the one that we generated. Yes. Is it OK to do overkill and just go to, like, X cubed first, just to make sure where you get it wrong? Even as ugly as that battle was, and by the way, that was a moderate battle for some of these problems, I don't think it's wise to continue to upgrade. I think it's wise to start with the one that you think is safe, because most of the time it is. And on a rare occasion, you're going to find out that it is not good. You're going to have to opt to the next solution. So what's next? 2A has got to be 4. So A is 2. We do get a solution. So let's go all the way back to our particular solution. We called it what? AX squared E to the 3X. We now know that A is 2. So it is 2X squared E to the 3X. So our final solution is the homogeneous solution. Let's go all the way back to that. C1 E to the 3X plus C2X E to the 3X. And then we'll add in our particular solution. That's probably a middle of the road kind of problem. Questions about any part of that? All right, let's start the next one. We've got five minutes. I know we're not going to finish it, but let's get a good start. And that way we will have seen an exponential, actually two different kinds of exponentials, one where we had to modify the solution. And then this one we'll get a look at one that has trig and not just any trig, but sine or cosine on the right side. Now, if the left side looks familiar, we have actually done this problem as an example problem. So this category of stuff on the right side will be a sine of something, X2X5X, or a cosine of something, or a sine plus a cosine. In any of those three categories, we would always start with the same type of particular solution. Characteristic equation, not going to factor. Quadratic formula, negative B plus or minus B squared, minus 4 times a times c, all over twice a. 100 minus 164 would be negative 64. Square root of negative 64. Heat i, thank you. 5 plus or minus 4i. So alpha is 5, beta is 4. So what is our homogeneous solution? That's the third case. So we will have reviewed the three cases as far as the homogeneous solution is concerned. By the way, we're actually doing this. Probably should write that down. That's what we're solving in part one of this problem. What is it? Mercy one, cosine 4X plus C2. OK, that's old stuff. Let's at least set this up, and then we'll pick it up from this point tomorrow. Particular solution. Well, we want to generate some sine of x's. Could we start with sine of x's with the original y thing, the first derivative of y, and the second derivative of y, and hope to generate some sine of x's? We're also going to generate what? Some cosines with that, because the first derivative of sines are cosines. Then the second derivatives of sines are back to sines. Does that work? So if we're going to generate some cosines, we need a way to get rid of the cosines as well. So let's start with some sine of x's. By the way, are there any sine of x's in any part of the homogeneous solution? We have some cosine of 4x's, right? And some sine of 4x's, but we don't have any sine of x's. Here's the kind of odd thing, is if we're going to generate sines, in this case sine of x, not only do we want to start with some sine of x's, we also want to start with some cosine of x's. And you'll see what happens when we throw the original y things in with the first derivatives along with the second derivatives. You'll see the interaction of sines and cosines. And it does become believable that we could just end up with sines, if that were the case. This would be a good place to stop, plus we're out of time. Another good reason to stop. See you tomorrow.


My name is Michael Dos Santos and I'm a commercial content producer. We produce photography and video for advertising. Running for me is definitely a stress reliever when it comes to my work life. It definitely, you know, whether I'm working in Texas and Japan, London, anywhere in the world, I can run. I mean, I grew up always loving sports and then played a lot of sports in high school, played soccer for a year, played football for a couple years, ran track for two years. More as a sprinter but I didn't really get into long distance running until my later years in college, just being from Boston. Running the Boston Marathon's always been, you know, a lifelong dream. Just growing up watching it on Patriots Day, whether it's on TV or when I was going to college in Boston, you know, going down there, kind of enjoying some of the party and the festivities around it. It's a pretty cherished marathon. It's a tough marathon. I'm excited to do it just to fulfill my dream and, you know, run the Boston Marathon because I'm a hometown guy. Representing the Pedro Martinez Foundation for me, I'm honored to represent the foundation. For me, Pedro, you know, he's a Boston hero, man. I love the Red Sox. I love Pedro. I watch Pedro. I was at the game, you know, he struck out 19 people like I follow Pedro forever. So I really love what him and his wife are doing with their charity and, you know, I thought it was a perfect fit for me to run the marathon for them.  Getting to meet Pedro before the race and stuff like that is, you know, that's exciting for me, exciting to meet someone like that.


My favorite Purdue tradition would have to be tailgating in the fall for football games, getting to see Purdue alum and current Purdue students all like mixed together and get ready to support our boiler football. I think that's a super important part of Purdue, and I think it's just a really good sense of community. My favorite part about my major would have to be all the clubs that Crantern offers and all the opportunities that they do with the career fairs and to get us involved with other companies outside of Purdue and also all the resources that they offer to just bring us together and promote Purdue. The number one reason I chose Purdue was how close it was to home. I after transferring I was ready to be a little bit closer to where I'm from because of family and I also just love the mission that Purdue has and all the opportunities that they give us.


Quickly, quickly, quickly, quickly, good. Well done, okay. All right, now your time is against you. There are more defenders coming back, so you have to play quickly. No extra touches, every touch you have is time for him to get back. So no extra touches that you don't need, okay. As soon as you can, you try and make that run in there. You get in the position where we can try and play you forward and the players here break. So can I make sure then on it coming, that first touch there takes me into my stride really quickly. And now I'm starting to go and attack people or attack the space to draw people together. Good, well done. Well done, well done, well done. Drive on, drive on, drive on, get high. Fantastic. Okay, now hold it there. That is terrific. That's terrific, that's terrific as well. Well done, leaving that, keeping the weight. Now what might happen is they might come over a little bit further. Now you might get your chance. It doesn't matter who scores. Well done in there because it might well be that you're getting slid in. A really good run for you is to run inside the fullback because you might just get slipped in. But that's a great touch, first of all, because immediately you got going forward. Ball out your feet, going forward quickly. And well done, what a good position. What a good position and good finish. Good, ready, play.


Typically, the thing that gets me excited about an MLB DFS slate is good pitching because I feel good about my ability to identify pitching. Typically it's my strength with regards to me as a DFS player. But today, I'm actually excited about the stacking options. It's a 12 game slate and I could legitimately talk up six different stacks and feel good about them and I think all of them have paths to really high upsides. So a bit of an inversion from the typical way I get jazzed about a slate and I think that should lead to a lot of fun. We're going to break down what those stacks are, how I'm ranking them, why they may be high upside or maybe risky, why it's worth that risk and much more to get you ready for Friday night. Welcome on into the solo shop. That's right here on the Fandual Podcast Network and Numberfire.com. My name is Jim Sonnis. I am a senior writer and analyst for Numberfire. You're going to break down Friday's 12 game main slate with lock set for 7 0 5 PM Eastern for today. The load of weather note on this slate, there is a high chance of rain tonight in St. Louis for the Dodgers and the Cardinals. They may be able to play through it, but I would check back on that later. There is an offense alike in that game. So we'll check back on the timing of the rain in St. Louis, the severity of it to see if they can get that game in. We'll talk about the stacking options and pitching in just one second. The first day reminder to make sure you're subscribed to the Numberfire Daily Fantasy podcast feed.  You name it. You can find us there. The solo shot also goes up over on the Fandual YouTube page. If you like what you hear, leave us a thumbs up over on YouTube or leave us a five star rating over on Apple podcasts. The second leg of horse racing is big three is here and Fandual is the best place to bet the Preakness stakes because right now all customers can get a no sweat Preakness bet up to $20. That means you'll get up to $20 back if your win bet doesn't win. Bet the Preakness with America's number one sports book. Just visit racing.Fandual.com for your chance to get a no sweat Preakness bet up to $20 this Saturday. That's racing.Fandual.com. Age and residency restrictions apply offer valid on first win wager refund issued in non-littrable racing site credit that expires on June 12th 2023 for restrictions applies to terms that are racing.Fandual.com gambling problem call 1-800-GAMBLER. Pitching preview for this Friday main slate Joe Ryan checks in with the highest salary on Fandual. His salary is $11,000 followed by Sandiel Contra at 10-4. We've got Bryce Elder and Bryce Miller squaring off elder salary 10-2, Miller salary 99. Marcus Strowman comes in at 95 with Tony Gonslin and potentially that rain game in St. Louis 93. Anthony DiSlaffani facing all Contra at 91. Then we have Blake Snell, Yusei Kikuchi, Ranger Suarez, Michael Kopeck, Martin Perez, James Paxton, Jake Irvin, Reed Deppmers and Kyle Gibson as the others at $8,000 are higher. Now of the guys in that list, the only guy pitching like a true ace so far this year is Joe Ryan and I think it makes a lot of sense to put Joe Ryan atop our list for tonight. Ryan is facing the Angels who have a 104 WRC plus against righties but they're not a low strikeout team, roughly averaged there at 23% and not a lot of walks either. The walk aspect doesn't really matter for Ryan much because he has just a 4% walk rate this year and it comes to the 30% strikeout rate. Ryan debuted a new splitter at the beginning of this year and clearly things going great with that new pitch. It does let up some hard contact but that has not killed Ryan as of yet. Both his actual ERA and his expected ERA are well under three. The twins also are starting to let Ryan go deeper in games. That's kind of been a trademark of Rocco Baldelli is taking pitchers out pretty early but with Ryan he's gone 100 plus pitches in three of his past five stars. He's gone at least six innings in every game so far this year. So I don't see a lot of reason to nitpick Joe Ryan. I think that he kind of just checks every box and makes a lot of sense. So to me, you just use Joe Ryan, feel good about that and proceed from there with your pitching options. Now I said Joe Ryan is the only one on this this late pitching like an ace. Sandiel Contra is an ace but he hasn't always looked like that so far this year. The bad at bulbate on him, not as good as it was last year when he was so dominant but he's still good and I think he could be a fun play tonight. He gets the Giants. They are a good offense against righties but they will strike out 24% strikeout rate against righties on the current active roster. That gives a boost to Alcantra and he can also get there in terms of DFS paying off his salary via volume across the path three starts here are Alcantra's pitch counts 103, 113 and 106. So as a result, I have Alcantra at the highest projected pitch count on the slate and that does matter a lot. Plus, Alcantra is still getting whiffs. He has a 13.7% swinging strike rate so far this year. It's just not always translating to strikeouts. That could change tonight against this high strikeout team. I have Alcantra projected for 6.7 strikeouts which ranks second behind just Joe Ryan. So if you want to buy low on Alcantra, I think this could be a good spot to do so. You're buying low on a guy with the second highest salary at 10.4 on the road against a decent offense. You're not buying low-low but I still think that this is a, from a rostering perspective, a spot to get Alcantra when he might not be at his peak popularity. On a lot of slates, I'm not super enthused by the value plays a picture because again values are typically values for a reason but this slate tonight is a bit different. Debtmer's salary is $8,100 and he's not perfect but I absolutely like him enough to use him for tonight. Debtmer's is facing the Twins who are a plus matchup for a lefty right now. They have an 88 WRC plus with a 28% strikeout rate which is the highest number on the slate. The strikeout portion for Debtmer's has been there this year. He has a 26% strikeout rate. His swinging strike rate is validating that strikeout rate at 13%. So the strikeout aspect of Debtmer's is a very easy sell. The reason that Debtmer's is lower salary right now is because of hard contact. He has led up a 43% of hard hit rate this year with a 40% fly ball rate and that's gotten him into trouble at times. It also could get him into trouble here because the Twins have some dangerous batters who can punish you if you make mistakes. But if you get strikeouts it may not matter as much and I do think that Debtmer's will do that. I have Debtmer's projected for a 6.6 strikeout tonight that ranks third on the slate behind Just, Ryan and Alcantra and looking at the salary you save a lot by dipping down to Debtmer's versus those two guys. So I'm not going to put Debtmer's above them because the batter ball data does matter and they are better objectively. But after accounting for salary I do think Debtmer's is at least in their tier tonight. That's good enough for me to use Debtmer's on this slate. I am more than happy to put Debtmer's in my player pool, see what happens, take the savings because I think he does enough to justify that on this slate. So to me, top pitching options for tonight will go Joe Ryan won, Sandi Alcantra two, Reed Debtmer's three and all three of those guys very much in play and well worth their salaries on Friday night. Now let's talk about those fun stacks and dig into the ones that stand out. Carl Kaufman coming up to make his debut tonight for the Rockies and luckily for him it's his debut but it's not a course field so that's good. The problem is he's facing the Rangers so it'll be a very powerful offense and I think the Rangers make sense from a stacking perspective here. Kaufman has been in AAA thus far and you can't really judge a Rockies pitcher ERA in AAA because of the park but the plate discipline numbers are relevant down there and Kaufman did struggle there as well. He had a 15% strikeout rates with a 7% walk rates. He let up a lot of balls in play with those two numbers combined.  We saw the same thing last year as far as a plate discipline numbers. The ground ball rate was a bit higher for Kaufman then but we're going to see a lot of balls in play here. He's probably not going to be super overwhelming from a ground ball perspective so I think it makes a lot of sense to target Kaufman here. We'll be stacking against him at Coors Field for sure but on this slate when he's facing the Rangers and a powerful offense I do think even here it makes a lot of sense to go at the Rangers as our top stack even on a really quality slate for stacking. The Rangers just got Corey Seeger back. Didn't do a ton in his rehab stint in large part because it was a very short rehab stint. Seeger came back a couple of nights ago and actually did have a couple of hard hit balls so I think that was encouraging despite not getting a hit. Two hard hit balls there. His injury slash illness is not like a red flag kind of thing for me so I'm fine being in on Seeger pretty much right away now that he's back. His salary $3,000, great. I think that's going to mean he'll be pretty popular for tonight but still makes a lot of sense. So sometimes with injuries you want to hold off for buying back in. I'm not super inclined to do so with Seeger. I think it makes a lot of sense to be on him from the jump. Number two stack is the Astros and they're facing a lefty and that's a big part of why I feel okay stacking them tonight because they are a very different team against lefties than righties. Against righties they lack power which means they lack upside. Against lefties the RISOs 171 which is much more stackable. They're facing Ken Waldichuk and the A's bullpen tonight so I think we should be high on the Astros here. The bullpen to the A's credit has been getting better so that's not as big of a factor anymore. I called them the Coorsfield effect before. That's not as true anymore but it's still definitely stackable. But Waldichuk himself is struggling. Across eight starts he has a 5.37 skill interactive ERA. He's walking too many guys letting up a lot of fly balls. His hard hit rate allowed is not a huge red flag but it's also not low enough to counteract the other issues in his profile. When you put that all in a blender it's led to a 7.02 ERA for Waldichuk this year. His expected ERA is 6.08. So we'd be looking to stack against him with most teams. I'm pretty down on the Astros but I think that they are still good enough to fit into this discussion. So I'm gonna put the Astros second behind the Rangers for tonight. Now when I'm doing that and stacking the Astros the one guy I will not be including in those stacks is Jose Ebreu. The numbers for him are pretty bad this year. He has a 265 expected Woba with 043 ISO and literally zero home runs. Maybe this is the night. Maybe Jose Ebreu finally breaks out of this slump that's been lasting for I think like over a year now. Maybe he finally gets it done. He double dongs gets back on track. Possible, but I'm okay missing out. I am okay not being there when that happens. I will buy back in once I see some sort of signs of life. I'm not convinced right now that Ebreu can get me a 30 point game on Fandul. And that's kind of what I want even from a value play like at least a path of 30 I'm not sure if he can get there.  That's okay. I'm very comfortable with that. I will be looking elsewhere within my Astros stacks for tonight. Now those are the two stacks that make a lot of sense from a traditional stacking perspective. The reason the slate is so fun is that you can talk yourself into a lot of other stacks where the upside is big. It might be risky, but that's an okay trade off given the upside they present. The two best examples of this are the Orioles and the Royals. I'm gonna talk about the Orioles here because they're a better offense, but the Royals will discuss some things to watch along with two other teams that great at well from a stacking perspective. The Orioles are facing Yusei Kikuchi who seems to have turned things around in some senses. Across eight starts he has a 3.93 skill interactive ERA. He's not walking a lot of guys and his ERA is good as well at 3.89. That's why it's not a perfect stack. There was risk in this profile for sure. The problem is that the bad at ball data for Kikuchi is still very rough. He has led up a 48% hard hit rate with a 43% fly ball rate. That's why his expected ERA is 5.03 despite the good results and the good skill interactive ERA. When you're facing bad teams, you can get by with that. The Orioles I think are a good team. I think they're legitimately fun. They have a 126 WRC plus against lefties, a 191 ISO. They put the ball near a ton. I think they're kind of a bad match up for Kikuchi. Now, because he does have some good plate discipline numbers and has good results, it's possibly could mow through them or they could capitalize on the hard contact and make him pay and get us some upside for DFS. I want to stack the Orioles right now in case they do the latter there because I think that's very much within the range of outcomes. Orioles, a great tournament stack for tonight despite the risks they come with. Kind of seems like Anthony Santander is getting back into his groove if you are a longtime listener of the solo shot. You know, I've loved Santander for a long time and he's back in the month of May. Santander has a 306 ISO. He has a 52% hard hit rate with a 17% barrel rate. That's all while reducing his strikeout rates not as low as it was last year but better than it was earlier on this year. The salary for Santander, $3,000. I love him as a one-off. I love him within the stack. He's probably one of the first guys to turn to within the stack for tonight. So we know how good he can be. He's slashing it against Anthony Santander fully back on the menu, $3,000, way too low for a guy with his upside and skill set. Things to watch, we're gonna go through three more spots we could consider stacking for tonight because that's how good the stacking slate is. As mentioned, the other risky stack with some batted ball upside is the Royals. They're facing Michael Kopeck who is a talented pitcher but and that's why there's risk here and that's also risky because the Royals are struggling but his batted ball numbers both last year and this year are rough. 51% hard hit rate allowed this year, 49% fly ball rate. The Royals are just an 80 WRC plus against varieties which means we can't get overconfident here but I think there's enough here for it to be viable. So I think both the Orioles and the Royals make a ton of sense and if you want just like Dinger potential I think those are two prime spots to get it. Hey, Tint, we'll talk about that later on. It's a similar thought process with the A's with the Royals where the A's are facing Brandon Bilak who had fine numbers in AAA. His first few appearances and the majors have been fine too but from what I also don't mind the A's they've got some guys who are palatable for daily fantasy. You could do a lot worse there. So don't mind the A's filtering them in as well from a stacking perspective. Finally, if we get the all clear on the weather for tonight, the Dodgers are facing Stephen Matz. Matz lets up hard contact which broadly means we can target him. The Dodgers, you look at them this year against left E's, it kind of looks like they're struggling. Their WRC plus is just 94. I think that number is very misleading. Their ISO is 212 with a 46% fly ball rate and those numbers are very good and those numbers stabilize much more quickly than WRC plus does. So I think if we expand the sample and give the Dodgers a full year they're going to wind up being a good team against left E's in 2023. If you get the sense they'll fly into the radar whether it be because of weather or because they're on the road, maybe people are high in Matz, anything like that. I don't know why, you know, something like that. Like that could be anything leading to there being a bit overlooked, I think is attractive. They're implied to be not super, super high. So I would keep an eye on the Dodgers, see if maybe we can get them at a slight roster rate discount. If we can, I'd be very okay going there and making them a priority stack for tonight. Dinger calls for today, mentioned before that the Orioles and the Royals are in prime spots for hitting home runs. So we're going to focus on those two teams for the Dinger calls for today. The boring one, Anthony Santander. He went deep yesterday. I think he goes deep once again tonight. Facing Kikuchi, he's just back. Looking at the main numbers, the bad-of-ball data, it's all good again. Facing Kikuchi lets up a lot of hard contacts. Santander hits well from the right side of the plate. So Anthony Santander, the boring home run call. The fun prize pass Quintino doesn't have more home runs given that he does have pretty good, you know, like a pretty good ISO against righties. But pass Quintino only has eight home runs so far this year. I think that's a decently low number for the potentially has a number of fly balls he hits and stuff like that. So we're going to put him in the Dinger call section for today facing off with Kopeck. So Dinger calls for tonight, Anthony Santander and Vinny, pass Quintino. That is all that we have here for today and this week here on these solo shots. I want to thank you as always for tuning in across this week. We'll be back with you once again next week, Monday through Friday, breaking down the MLB DFS Main Slates, also talking PGA and USC for select events. To get those as they are posted, make sure you're subscribed to the Number Fire Daily Fantasy podcast feed wherever you get your podcasts. If you've got any questions for me, I am on Twitter at Jim Sonnis, J-I-M-S-A-N-N-E-S. Hope you all have fantastic weekends. Good luck to your MLB DFS line of search tonight. We'll talk to you once again Monday for some more MLB DFS. This has been the solo shot right here on the FanDuel podcast network.


Did you know that there are over 2,200 known religions in the world? We are all born into one belief, religion, or faith, which influences how we see the world and everything and everyone in it, including ourselves. Do our beliefs divide and separate us, or do they bring us together in greater harmony? When you look up with awe on a star-filled night, do you ask who or what created all of this? Have you ever had a profound or deeply challenging experience in your life that changed your beliefs at the core of your being? Enlightened Pathways takes us on a journey of discovery to understand just how spiritual transformational experiences impact our lives and the world around us. Join us now as we deeply explore all that nourishes, heals, and inspires us. Welcome to Enlightened Pathways. My name is Robert Quebec, and I am your host for today's spiritual journey. Our special guest today is Henry Jampolsky. I became fascinated with Henry's passion for his spiritual practice and thirst for greater knowledge and awareness. Many years ago, when we first met in about 2017, I think it was at a business networking event in Warrington, Pennsylvania, and we quickly became colleagues and friends. Henry's spiritual journey even took him to the highest pass, a treacherous and arduous motorcycle journey through the Himalayas that awakened his spirit in profound ways. And Henry is an author, a mediator, educator, TEDx speaker, and who serves as the Assistant Director for Education, Outreach, and Conflict Resolution at Virginia Tech's Office for Equity and Accessibility. Henry is also a master instructor of Satva Yoga, having studied in Rishikesh, India, which is a foundational inspiration for his new and best-selling book, Dissolving Conflict from Within. So please help me welcome Henry Jampolsky. Thank you so much, Robert.  And thanks for making time for us today, too. Can you briefly tell us where you grew up and what religion or belief system or faith that you're born into and what impact that had on your early life as you recall it? So I was born and grew up and spent my most perhaps formative years in Ukraine. And I was born Jewish. Though my experience of being Jewish was not religious necessarily, but actually being Jewish in Ukraine was more of an ethnicity. And at the time when I was growing up, which was in the 80s and 1990s, there was a great deal of discrimination towards Jews. And so for me, from a very early age, this was something that was othering me and in many ways was closing off many opportunities that were available to me. So I did not really have much of a religious practice growing up, but just being Jewish and that being considered at that time a so-called less desirable ethnicity and something in many ways that you were defined by in many ways define my early years. Now, when I came to the United States in 1994, we actually came as Jewish refugees to escape anti-Semitism then prevalent in Ukraine. For some time, I did become more religious going to the synagogue and connecting, because for the first time in my life, for me and my parents and my members of my family, we could practice this religion unimpeded and that was very meaningful. And so for a while, I became more religious and attended services. And when I was 18, I went to Israel. And if we're coming back from Israel for a while, I felt even more connected to Judaism and being Jewish as a religion. Though it's time progressed, I moved away from that. And what was it that you think caused you to move away from that? It was a... I don't think there was a single event that caused me to move away. For me, it was more questioning. I grew up with a questioning mind. And the more I was observing life and connecting and growing up, the less I felt connected with something that was written in a book many thousands of years ago and something that very often I found to be inconsistent and something that seemed way too binary to me. And so I slowly started moving away from it. Then I met my wife and when we met, even though she also was Jewish, but she was not practicing at all, we also fairly quickly and at a fairly young age, began exploring spirituality versus religion. And we felt much more connected with spirituality than with any particular religion. And then, of course, eventually I ended up in India and that in many ways turned all of my... everything that I thought was up became down and everything that I thought was down became up. And that really transformed how I see myself, how I see the world and what it is that encompasses my spiritual practice. And I really love how you explained that, Henry. Thank you. And as you know, I have been to Israel many times, Palestinian, Lebanese background, lots of family there and being in Jerusalem and Nazareth and what have you. By the time I went there as an adult, you know, I also had left my religious upbringing as it were and I was much more fascinated with the components of it. So, and like you, it sounds I started to look for spiritual common denominators across a number of these practices because, you know, one religion made me feel part of it. And at the same time, another felt made me feel out of it. And this practice allowed me to do this, but then it didn't allow me to do that. And I don't know, I find myself seeking more flexibility in my life, which religions do provide. In a lot of ways, it just wasn't something that connected with me. I'm very interested though, can you talk more about your experience going to India and how things that were up became upside down as a word? Sure. So first, let me let me tell you where I was as a person when I went to India. You know, I was coming here as a refugee, coming here as an immigrant. My family had to go through a lot. It was very challenging, but in speaking English, we had to really advocate for ourselves and did not have the skills to do so. So I thought if I become a lawyer, I would be able to be the voice for people like my parents, for immigrants and other people who did not have a whole lot of power. So that's what I did. I became a lawyer. It was very successful at it. I did a lot of civil rights work and other types of key internal, the other types of cases where I represented people who are mostly poor and many were immigrants and many did not have much of a voice in American courts. And the more I did that, the more disillusioned I became. And the more clear it became to me that I was not providing the voice for the people. But in fact, creating more and more separation, creating more and more conflict. And I was beginning to become very concerned about what this work was bringing out in me, that it was bringing out the worst in me, what it was bringing out in my clients, and what it was bringing out in other people that I came across. And this is when in seven to eight years into my legal career, I felt really lost. I was very successful at it. I invested a lot of time, money, energy to become a lawyer. And yet this work was not fulfilling for me. And I wanted something different. I wanted something different in life. And this is when motorcycles entered into my life. And I really was just looking for a passion. I really was just looking for a passion and just something to be thrilled about because I wasn't at the time. And so I started riding motorcycles. And this was not something that was easy for me. And then one day, I just was not feeling well. I felt down. And I was just playing on Amazon. And I come across this movie called The Highest Bass. And this is a film about a young Indian mystic who takes a group of westerners on a motorcycle journey across the Himalayas. And when I saw this film, I was just blown away. And I said to my wife, you know, I got to meet this guy. I got to meet this guy. He's crazy. I got to meet him. And so we looked him up and he was actually teaching a workshop in Virginia. And we were in Pennsylvania within driving distance. So we went and spent some time with him. And, you know, I've never encountered a being like that. I've never encountered a being who was so innocent. One moment and could be so serious and profound the next. And did this so effortlessly.  And he said to me, you got to come to India. And then maybe you just won't be a series, you know. And I never thought that I would go to India. Of course, I never felt a particular connection with India as opposed to any other country in the world. But after meeting Anand, I felt very driven to go. And so a few months later, we took I took a month off from work, and I fully expected my work to tell me to take the rest of my life off, which I ended up doing later. But at that time, I wasn't ready to make the move. But my work was very, very supportive. And we ended up in India. And I can tell you the biggest difference and the biggest lesson as to what India brought for me and being there.   And here for the first time, I got connected with a spiritual tradition that told me to look within. And told me that the answers that I was seeking are not going to be found in a book. They're not going to be found in some building. They're not going to be found through a particular ritual. That until and unless I was willing to go within and sit with the unpleasant and sit with the ugly and experience the full range of emotions and experiences. Nothing was going to happen. Nothing was going to happen. And then something else that became very, very clear to me was actually it's not about any kind of set of beliefs. You know, Robert, I know you have some experience in chocolate. So I'll ask you, do you believe in chocolate? I believe in chocolate. Well, but let me, let me maybe correct you a little bit. You actually don't need to believe in chocolate because you know chocolate, right? I know you've been a chocolate chef. Yeah, you know chocolate, right? I know chocolate. Yeah. Right. You know chocolate. So you don't need to believe in chocolate. This is something you know. This is something you've experienced. So when we say I believe in this or I believe in that, whatever it is, we are talking about something that we have not experienced because once we experience something and we know it, we don't need to believe in it. And so fundamentally what also became very, very clear to me is, you know, up to that point, my exposure to Judaism and other Judeo-Christian concepts was all in form of the belief. But it was when I came to India and by the way, I want to make it clear, going to India was not about connecting with any particular religious tradition. You know, even though now probably many of the Hindu practices are what I connect the closest with, I would not consider myself a Hindu or Buddhist or really anyone else because this is not to me about kind of labeling myself and naming and you know, I am this and not that. But fundamentally, fundamentally the teachings that I came in contact in India opened my eyes to something, to this and it was to experience, it's not about believing in divine or anything else, it's about experiencing it. It's about experiencing. And the only way to truly experience it is not to look somewhere out there, it's not even to look at the highest fast in the Himalayas, I tried. It's to go within. It's to go within. And as we start going within, as we start going within, some very, very powerful things begin to happen. And these powerful, the most powerful things is naturally we start to expand.  If you look actually at the description of enlightenment from masters like the Buddha and others, what they're describing, what they're describing is the experience where all of a sudden they couldn't or sometimes it's not all of a sudden, but they cannot tell anymore where they end and another being, another person begins. And so as we go more inward, as we go more inward, naturally our perception, our awareness begins to expand. And naturally we start becoming more inclusive. What does it mean to be inclusive? Inclusive means that I start actually perceiving and experiencing you as part of me. Now, when I start perceiving and experiencing you and anyone else and all the other beings as part of me, the integral part of me. So my idea of me expands. But then does anyone need to tell me not to kill you, not to harm you, not to do all these things, you know, so many religions became these manuals, right? All the things not to do, all the things you're not to do. But the moment you experience another being as truly part of you, all that falls away. Now, you don't need, you don't need these rules, you don't need these regulations.  Of course, if you take my arm and my leg, they look different, they function differently. So it's not that we are the same. And it's not that we forget about the differences. But if we see ourselves as part of the ocean, and then as ocean, the idea that I can harm you, the idea that you're different from me, the idea that you're other, other than me, drops. And then the violence drops. And that description that you just gave, I resonate with so much in the perspective of being able to experience through various forms of my own meditative practices and experiencing where that line doesn't exist between perceiving myself as one thing and perceiving spirit or awareness or consciousness as something else. Being able to look at a whole as opposed to separate. Yeah. And see, these ideas, if we look at many of the religious traditions, and of course, you know, religion can provide religion can provide a lot of solace, religion can can provide a man solace. There can be a sense of community. There can be, you know, immense strength in some of the rituals that can inspire us, for sure. But while religion can be a solace, it is not a solution. And there is a big difference in that because fundamentally many religious traditions in the world perpetrate three very dangerous beliefs. And these are beliefs because they're not based on experiences, but beliefs that become so deeply ingrained in everything that we do that they inform how we are. So one belief that we started to talk about is this idea of separation, right? That there is you and me, and that we are separate, and that somewhere out there, up there, up there, or down there, there is a God who is also separate from us, right? So this comes an idea of separation, and that, you know, if you are identified as this person, and you fold your hands in a particular way as you pray, well, you cannot be in the same room with someone who folds their hands a little bit differently. So this idea of separation, right? The second idea is that many religions, or at least what people take from that, right? Because that's not necessarily what the origins of the teaching start. That's not necessarily at all what the origins of the teaching start, but kind of the dogmatic interpretation is a idea of separation. B is that we start identifying with our accumulations. So this body is an accumulation, right? This mind is an accumulation. Everything really we're sharing is an accumulation. So while it is Indian mystic satguru often teaches, while it is ours, it is not us. But we confuse that. We confuse that all the time with what is ours and what is us. And that results in immense pain. That results in immense pain because we're constantly looking for solution somewhere out there. Yet the most we can find out there is maybe some solace, but never a solution. The final belief that in my view is very dangerous is that our lives are both infinite and transitory. Meaning that we're here just kind of passing through on some way to a better place. And so in a way what happens here and what we do here is just preparation for something else. So it's kind of like the rehearsal and it doesn't matter so much. And what that results in, right, is lives where, you know, in reality, and I think you and I are both now entering the age where, or perhaps enter the age where life lies, you know, we realize we're not here for that long.  Where have they gone? And where have they gone? And then you look and, you know, you say, we have limited time left. But we're living as though we have unlimited time, right? So this is where we're spending all this time scrolling and doing all this other nonsense that we do while life is sticking away. And then we feel disconnected from this very essence of life, right? Because life is not something that we're going to learn how to do in a book, or some coach is going to teach us. Life is something that we have to experience. And for me, you know, I moved away from Judaism, I moved away from Western based and Judeo-Christian based religious practices. Because fundamentally I saw the focus here was on these three beliefs that I saw not only not corresponding with my experience, but also being very harmful and inherently being very violent. And yeah, so let me ask you a question that keeps coming to my mind in the few minutes we have left here is the journey that you've been on to actually allow you to come to these conclusions where there are other specific sources that you were able to compile this information from. Or a lot of it sounds like a lot of it's just been internal discovery through that internal connection. Can you explore that just for a minute with us? Well, sure, there were other sources. There was my teacher, you know, Anand Moroch and there were many, many other teachers. You know, of course, there are books and spiritual books, though, if we look at the yoga tradition, I purposely don't call it Hindu tradition because, you know, Hinduism kind of evolved from that and also has its dogmatic elements. But, you know, one of the teachings, for example, the yoga sutras by Patanjali, it's a tax that predates Christianity by, I believe, 3000 years. And it's a series of 200 incomplete sentences, incomplete sentences. And yet, each sentence, each incomplete phrase, you know, their partial sentences, to me had a deep deeply profound immense meaning. So, for example, the first yoga sutra, now yoga, now yoga, right? It's very easy to dismiss this. It's very easy to dismiss this to say, well, okay, now he's going to talk about yoga. Except that's not what he's saying. There's a deep purpose behind it because if we want to experience life, it can only happen now.  Yoga meant unity. So if you want to experience unity with life, meaning you want to be one with life, that can only happen now. It's not tomorrow. It's not yesterday. And that can only happen through experience. It's not through by reading a book. So it immediately, even through this one sentence, there is an immediate invitation to go within. Another example I can give you is Bhagavad Gita. You know, Bhagavad Gita, of course, is an ancient yogic text that describes kind of one of these epic dialogues between Arjuna, the general and Krishna. And it's a yogic text about love. So when you say yogic text, you would think, okay, well, probably the teaching is going to be make love, not war, right? That's not at all what they're saying. So Arjuna is this warrior who's about to go to war actually with his cousins. That's kind of how the ancient India worked. And he kind of says, eh, I don't want to do this. I don't feel like it. And Krishna, which is an embodiment of the, who's the embodiment of the divine, says to him, your job is to be a warrior. Stop complaining. Your job is to be a warrior. He doesn't tell him your, you know, make love, not war. No, he says your job is to be a warrior. You need to do your job. But then he says something very, very profound. And he says, establish yourself in yoga. Establish yourself in yoga and then act from there. So go within first. Go within first. Establish that stability and then act from there and then take the appropriate action from there. And then if you're established within that action, it's not going to be dogmatic. You're going to know what to do. It's not going to be driven by this set of beliefs or that set of beliefs of what this person said on this TV channel or what that person said on this, on that TV channel. And so fundamentally, this became my spiritual practice going within, going within, going within. And you know, a very Western idea of meditation and mindfulness and yoga is we do this to feel better. But the invitation that I hope to extend to your listeners, your viewers, is to consider switching, to consider switching from doing this to feel better, to doing this to get better at feeling, to doing this to get better at feeling so that we become more sensitive to ourselves and what is happening within us. But more importantly, so that we become more sensitive to life because everything is life, us included. And if we're not something that a coach can command and teach us how to do, unless we are sensitive to this, unless we're really experiencing it, we're not really living. Yeah, thank you so much, Henry. Very much. Thank you. And that's a great place to end our conversation today. And we could talk a lot longer on this. And your depth of knowledge is constantly fascinating to me. And I thank you so much for your time. Again, Henry MPolsky here now, the bestselling author of his dissolving conflict from within. If you'd like more information about Henry and his workshops, his books, you can visit www.livingpeaceinstitute.com. Again, thank you, Henry, for your time today. And just a few closing remarks and a shout out to today's executive producer and sponsor, Bridge to Heaven Healing and Lipin Lizards, which is the premier source for healing crystals and readings with four locations. You can visit www.lipinlizards.biz for more information. Also, a big thanks to our co-executive producer, Dr. Anna Cabeca. Visit drannacabeca.com for more information. Also, if you would like to get more information about this show, please visit www.deepbeing.org. Thank you for watching Enlightened Pathways and spending your valuable time with us today. Until next time, play, have fun, be happy.


Tom here from Orange Systems. It's January of 2023, and we're going to talk about PF Sense boot environments. I figured it's a great topic because we have the release candidate here available as of the recording right now for PF Sense Plus, and I've been testing it out. And boot environments are a great way to test without the risk of having to reload your firewall if it doesn't work for you or if you find a bug with your configuration that, well, just makes you want to roll back to the previous version. And rolling back doesn't mean reloading the firewall, it just means selecting the previous boot environment. There's a couple of prerequisites. First one is ZFS. You have to have a PF Sense that has been loaded with ZFS. There's not an in-place conversion from UFS. So if you have a system that you would like to convert, the only way to do this is going to be to download your config file, reload the system, push your config file back, and it will restore with all of your settings, and then you'll have ZFS. There's no in-place just click button, convert here type of upgrade. Next, boot environments are only available for PF Sense Plus. Even though ZFS is supported since, well, it has been the default, I should say, since 2.6 on Community Edition and the 22 series of PF Sense Plus that was released in 2022, you only get the boot environments inside the PF Sense Plus Edition. If you'd like to get PF Sense Plus, it comes free with any Neckate device. You can get a free home user license, a free lab license, or you can get a business with support license all from subscriptions from Neckates. There's some other differences you get with PF Sense Plus. I have a separate video linked down below to dive into that topic. Now, the advantage of boot environments is that ability to just migrate between different versions very quickly, and that's what I wanna walk through. I also will leave a link to the PF Sense documentation, but we'll actually walk through the upgrade process of going from 22 to 23 and using a boot environment so we can revert back if this upgrade doesn't go well. This is my lab system that is running the 22.05 release. It is loaded with ZFS, so we have our boot environment option here. This is the 22.05 working edition, base version 22.05, and let's go ahead and clone it before we start messing with things because we're gonna do an upgrade to version 23. Tells you how much space it's using. Let's go ahead and snapshot this. So we'll call it 22.05 working. Before upgrade, seems like a good name, and you can give the same descriptive name, but descriptive names allow you to have spaces and be more descriptive about what this is. So we'll call this 22.05 working before YouTube demo. So hit save, now we have this one. It's only taken a few kilobytes because these are ZFS snapshots. They're points in time for the entire OS, not just the config here, but the entire operating system is snapshotted. They don't take up a lot of room unless there's a lot of changes in them. Now currently the little star here means that is the next boot environment as in the boot environment that will be booted when this system reboots. If I would like this system here to be rebooted, so we'll click this. Now this one will be the one on next reboot, but the green check mark here means the current one we are working from is this. Now the next options are gonna be obviously edit, you can rename existing ones, you can delete them, or you can activate a one-time reboot. So if I just wanted to change something in this and I could hit the play button here essentially and it would reboot one time to that, but then the next reboot would be back to this one because it doesn't set it as default. But pretty easy just to choose these back and forth. You can also quick click the quick create and it'll quickly just create a snapshot and give it a date. And once again, the same option pulls up here if I wanted to set this one to be the boot environment. But we're gonna ahead and delete that one. We only need this one here. Now let's go ahead and walk through an upgrade. So this system is working fine. I hope it works fine with the upgrade, but just in case it doesn't, that's why I made this working before upgrade one. So let's go ahead and do an update to this by going to system update. And we're gonna choose the 2301 beta here and hit confirm. And it's gonna go ahead and update the system to the new beta. We'll fast forward through the update process, but basically it's gonna download all the packages, apply the update and reboot. The good news is it looks like it pulled up the web interface. Let's see if we can log in, click accept, close. And we are now on the 231 beta. That's awesome. Hopefully everything works. But of course, this is about boot environment. So what if something doesn't work? It did create an auto snapshot or boot environment. Let's me know the base version for that. So the base version is the same because this is the one we created before upgrade just so I had a snapshot. And here is the one that was automatically created by the system update. Now let's suppose I wanted something back in 2205. Now any changes I'm making now are always going to the active version. And this is the one active of boot. But if I wanted to revert back to this one here, I just simply click the star and that will set this to be the boot environment that I wanna go back to. So we've set this and I'm gonna go ahead and restart the system. So we'll go to diagnostics and just reboot. And we can see the boot environment is going to be this one. This is a nice feature of the restart features. They've added it here as well. So you can see the different ones. Now it's probably worth noting and let's go back over to our boot environments here that I should rename these real quick because if I don't, I could create my own confusion. So I know this is 2301 release candidate. So we'll save that as this one's name just to remove any confusion. I see the base version here but you'll see later when I show where we do this from the command line it will only show these parts of the names. And it's important to have those clear so you don't get mixed up. But let's reboot it and show you what happens when we revert back. It's pretty simple here. So diagnostics reboot. We know it's going to go to the 2205 normal reboot hit submit, let the system restart. Now we're here at the console. We can see it's gone back to the 2205 release. We'll refresh the page, log back in. And here we are back at the 2205 just like nothing happened. Now any changes I make here are not going to reflect into 23 version and vice versa. It's always writing to whichever the active boot environment is. But if we wanted to go back and forth and maybe try this or maybe there's another update or some other configure we found a workaround because we want to go back to this. We can simply go here and go back to it. And if we wanted to export changes between this version and that version we could just take the diagnostic backup restore file and push it to this version by booting into that environment and loading it on there. But the next thing and let's go ahead and show you what happens on a reboot if you want to choose a environment from the console because well, your upgrade didn't go so well or you made a change that has now put you all the way back to not being able to get into the web environment here. So if you can't get to the web environment yes, you can do it from the console and let's walk you through that real quick. Actually we'll just leave this one at default and reboot it and just bring you to that menu so you can set the next boot environment. So from the console we're gonna just gonna choose a reboot system. Yes, reboot normally. Now from this menu we're gonna choose option eight boot environments. Now this is where it can be a little bit confusing. We have the active and we have the boot FS. So currently is active on 20205. And if we wanna choose a different one because it says one of three, we keep pressing two and it chooses a different one. This is the auto one here and this is a 23 version here. So it cycles through them. It does not list them. So it can be a little bit of a nuance. It's in the documentation but it's something that can be easily overlooked like, hey, what is it doing? And just to be clear you're cycling through the different options and the different boot environments you have. And this is why because it does not give the description it only gives the name of the boot environment having them clearly named like 2301 working RC for release candidate. Now we know this will be the active one and we just go one back to main menu. Now this is also letting you know what the ZFS route is the working before upgraded to one that would normally boot our override is what we see here in option two. So if we choose three you see it says the active one. And what three is actually doing it reverted number two back to what it's going to be. So if you cycle through them and maybe you had 10 of them you just want to get back to whichever one is the current boot pressing three we'll bring you back to that. Like I said, this is covered in documentation I just wanted to make it clear. So let's go ahead and boot back up into the 2301 environment. Choose that option, press one, press center and it's going to boot up into our 23 environment. So we're back in our 2301 beta go here to boot environments and I want to point out the 2205 is still the default when you do something from the boot menu option in the console it's only setting the temporary boot environment it is up to you to make it the permanent one by well simply clicking the star right here and now this one will be the boot environment that you will have on reboot. Pretty simple to do and easily I could have selected any one of these as needed. Now I think the boot environment feature is super cool. It allows me to easily take firewalls that I want to test or even ones I'm testing at home because maybe I want to try out the release candidate but I'm not sure if all the complexities of my system will work fully or maybe it will get me almost there and I'll find some bug and I want to be able to quickly just hey I just want to reboot and revert back and I don't want to have to go through any reload, revert process. Boot environments are great for doing that great for testing ideas and scenarios that if you want to make permanent you just change it to say hey I want to go ahead and make this one permanent and I can always delete that old boot environment but I think this is just great for hardware firewalls when you don't have them virtualized I know someone's going well just virtualize it because that solves the problem too solves the problem in a different way but I do really recommend it for many of our clients they are running hardware firewalls. Now one caveat here this is no substitute for a backup at all you should always grab a copy of your config.xml file out of the system because that is the backup because there's still unless you've set up ZFS RAID in your system which is still resiliency and not a backup you still are at risk to lose all of it if that drive dies even if there's boot slices on there as good as ZFS is it can't protect you from a major hardware failure so as always backup head over to the documentation for more details and leave your thoughts and comments down below head over to my forums for more in-depth discussion head over to NetGate's forums for a more in-depth discussion and questions about PSNs thanks and thank you for making it all the way to the end of this video if you've enjoyed the content please give us a thumbs up if you would like to see more content from this channel hit the subscribe button and the bell icon if you'd like to hire a short project head over to laurancesystems.com and click the hires button right at the top to help this channel out in other ways there's a join button here for YouTube and a Patreon page where your support is greatly appreciated for deals, discounts and offers check out our affiliate links in the description of all of our videos including a link to our shirt store where we have a wide variety of shirts that we sell and designs come out well randomly so check back frequently and finally our forums forums.laurancesystems.com is where you can have a more in-depth discussion about this video and other tech topics covered on this channel thanks again for watching and look forward to hearing from you