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In Exercises $1-24,$ find the general solution to the given Euler equation. Assume $x>0$ throughout.
$$x^{2} y^{\prime \prime}-3 x y^{\prime}+9 y=0$$

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Hello doing today? We're going to start from the number 22 he had given. Question is 16 square my number dash minus eight expired ish last night by equal zero here. A total 16. Be good to minus it. Seiko. Dona. So hey, all square plus B minus A door minus eight. Minus 16 into, uh, let's see, he called zero. We got 16 Heart square minus 24 are last night because it'll I could do trained by for Goma three by foot. So why off this exhibit B fever three by four is there. Let's see to their ego about off three before is there some students that they could go? Lennox? Why off ecstasy, Ricardo. See one people about three by four her legs plus c do Atlanta eggs. And do you know the bottle? Three. Before Olynyk begin. See? You are excellent, Bubba. Be my foot less futile. Hellenic X to the about off. Read my foot. Thank you.

All right. So we have d times the quantity 18 minus nine is equal to negative one. So, uh, we need to solve Fergie's. So the first thing I'm gonna do is I'm going to distribute the D onto both terms. So first, um, I have d charms 80. That was just gonna give me a t swear. Then when it goes on to the negative nine, that's just gonna give me negative nine. And so next, I'm going to bring that negative one over to the other side so that the right side of Merrick region is just zero. The reason I'm doing that is looking at this. I can see that I have a quadratic equation. And with quadratic equations, the best way to solve them is my factory. So, uh, I'm gonna add to earth, so I'm gonna add one to both sides, and that's not gonna change, um, the 80 squared or the negative 90. But I'm just gonna have plus one added onto the it. So I have eight p squared, minus 19 plus one. So quadratic expressions are normally in the formula X squared. Most York's Lucy. Um and I'm just using X in our case, Our variables. D That doesn't really matter, though. Um, we're A B and C are just numbers. So it's your factor. We use something called the A C method or we take our A and C and we multiply those and take our B. And what we're doing is we're looking for two numbers that will multiply to a C, and those same two numbers have to add up to be. So let's think about what our A, B and C or in our case, R A is H. Because that is the number in front of the D squared R V is negative. Nine. That's number one of the tea in our C is one because that is the number that doesn't have a variable. So 80 times see, in our case is just gonna be eight times one, which is just eight. An RV is negative nine. So we want to numbers that multiply to make eight, and they add up tonight. So the the signs actually give me a few clues here. It says the ADA is positive. That means I have multiplied two numbers to make a positive number. That means either of those two numbers were both positive, or those two numbers were both negative. But if you think about the rules with multiplication with negatives, um, when you multiply, two positives will get a positive. And when you multiply two negatives, you'll get a positive. What if you multiply a positive with a negative? You will not get a positive. So know that Archie Numbers have ties where they have to both be negative, then our next clue is that they add up to a negative number, so full of the more numbers were positive, that wouldn't be true. That wouldn't happen. So we know that I were two numbers after both negative. So that's great. The next thing, when you do is think about the factors of eight, because those are the only things that multiplier to make. Eight. So there's one turns eight. There's two terms or and that's all there. Really, it's so we know that both numbers have to be negative, so we have to check those negative one minus a equal negative nine, or does nephew to chew minus for equal negative night? Well, the answer is it's gonna be negative warm, minus eight. Uh, because that is equal to negative nine. So great, we found our two numbers. We know that they multiply to make positivity and know that day. I, uh should make negative night. So perfect. Now, what we're gonna do with those two numbers is we're going to split our be on little term into those two numbers into the some of those two numbers. So negative 80 minus D and I'm gonna bring the rest of the regime down. So the next thing we're gonna do is route the 1st 2 terms, and then we're gonna group the last two terms, and I'm gonna put a plus in the middle just so that we know these are being added to get it. So I've grouped these terms so that I can see if I can factor anything out out of the first grouping. They both have an eight so long in a factory on E. And they both have at least one deep. So I'm gonna factor out deep, and that's gonna leave me. Were deep, um, minus one out of the second term. I have a negative sign in front of the D and I'd rather not have a negative sign on my first trip. So what I'm gonna do is I'm going to factor out a negative one, and then that's gonna give me a D minus one, because there's nothing else we can manufacture out of there on. The reasons that this is d minus one is because factoring up the negative one is kind of takes the negative out of the D. And it changes this positive warn, injuring negative one soon. And we have term up here and then a term over here, separated by that plus sign. I have two different turns in both terms have a D minus one being multiplied to them. So what I can do is I can factor out the D minus one the way you would factor out a number. So I factored that out. And on the first term, I'm just left with eight D. And on the second term, I'm just left with the plus negative one. And instead of breaking glass, negative one, it might be a bit easier to just write minus four, because both things very mean the same thing. So now we have completely factored or or driving equation, and now we have to think about something called zero product property. So there's your product. Property tells us that if I have two Numbers X and why and I multiply them and that gives me zero, then that means either X zero or why was zero for both Ex Admirer zero? Because that's the only way you can multiply two numbers to get zero. At least one of them have to be zero because there's no way to multiply random numbers together to gets your It's not possible. So that's kind of what's happening here. If you think about it, we have a number being multiplied to another number and equal zero. So either that first number Do you mind this one is zero or that second number 80 minus one is zero. Then all we have to do with soul, both of these for a d and then that will tell us that's how we have sold for geek. Pretty much so on the first equation, I'm going to have one to both sides that's gonna give me d equals one. Um, is these cancel? Oh, and then I have zero plus one on the right side Looking on the left side. I'm gonna add one to both sides as well. These were gonna cancel. And I have 18 is equal to zero. Plus one is just one. And you get the deed by itself. I'm gonna divide by eight, because right now it's in multiplied. So to get rid of eight, I have to divide. That's gonna cancel. And then I have to do by the other side by eight of 12 and I'll find that he is equal to what? So then my two solutions RG Cole's one. Indeed, this morning.

We lost the focus all those 28 Multiplication More Low eight two. In fact this can be sold in two methods. Well understand both methods and we will know which method will be the best according to the given problem. So in the first method, what we do is we take both numbers that is 28 and to multiply them together. It is 28. multiplied with two will queue 56. Since we deal with the multiplication Modelo eight, we have to divide this boy eight and find the reminder. So let's do that. So when we divide of 56 x eight We'll be getting seven as a caution which means eight times 7 is 46th and the remainder is so you know, so we get the remainder at zero. So therefore the answer for this question is zero. Now let's see this again method. In the second matter what we do with we look out for any bigger number which is Greater than eight. So first we More out. So first we divide this by eight. Since this is 28. Since the 28 is greater than eight we can more this out With eight. And when we do this we will divide this by eight We will have 383 times since 24. And then the reminder is for So now this remainder four Is less than eight. So what we do is we we can replace this 28 as four multiplication. More low aid To which means now this is equal to four times 2 and this is equal to eight. So finally we have to mark this out with eight, which means we have to divide this eight birth eight And we'll be getting the remainder zero. So in both cases we get the remainder as zero. So let's now understand which method is best for the growing problem, particularly when the number is skated. Then the maud M, the more of the M. Then it is always best to go with this appointment. In case if it is less than if both numbers are less than the more of the particular number. That is, if you see the M or M. If both numbers are less than M, then it is always supposed to go with this method. The first method.