Y"" Ty" + lly 5y = 0 with y (0) = =l y (0) = 1, and %"' (0) = 0...

$$y^{\prime \prime}-y^{\prime}-2 y=0 ; \quad y(0)=-2, \quad y^{\prime}(0)=5$$

Now why Double prime? Let's to y prime was five like, Well, zero is our problem here. So the characteristic equation is R squared, plus two are it's five equals zero, and that implies that are is equal to native one. Let's remind this to I. So why, If he's going to be given us, see one times, it's the negativity Times Co sign of to t plus C two times each. The negative t then sign to t. This is our final solution.

Ah, good day, ladies and gentlemen. We're looking at a question number 23 here and involve solving the initial now your problem as given here and you'll notice that, um the form of initial are the form of the non. How much in your sick equation, um, allows us to use the method of undefined, cool, efficient or under terminal missions? And first stop in that method is to approach the genius equation be induced homogeneous equation, which in this case, is why prime minus white will zero, which has the auxiliary equation, are minus one and whose roots is our equal one. So from there we get the general form. Are the general solution of the homogeneous equation being C one each of the tea? And now from here, um, you'll notice Okay, since the, um I guess the little af of tea, which in this case is is this guy which is one is not a linear combination. Off is the homo genius equation. So we can we can to solve the homogeneous equation. We could just use the form of the gas, which is just why peak was a because one is not a linear combination off e to the T or you know, is not a linear combination of of I guess this is why h sorry about that. I call this why h here. Since it's not a linear combination. Why age? Um, we can then just take our guest to be a which is the highest degree of the polynomial here. Um, and then we input it. And, um uh, why? Why Prime To get negative A is one or just Of course. Um why, Peter b. Negative one. Okay, so now we have r Y P. We have our y h. And of course, And we can Now we can go about solving the, um we have the general form. Um, sorry. We have the general form, the solution to the non homogeneous equation, which, of course, is just why age plus y p here and now, use that to solve the initial value problem. So, um, you plug, use this and you, Plus you using it, you use Sorry use. Why here and the initial conditions here to get another system linear system of equations. In this case, it's just gonna be ah, why of zero, which is just see one minus one equals zero or see one equals one. Okay. And then that tells us that the solution of initial value problem is just weikel speed of the T minus one. Okay, so, um, as you can see, uh, now the problems are going to start getting longer and solutions when you want to solve the actual value problems, because you're gonna have to have this added step of the non homogeneous part which gonna make it even longer. So, um, but still pretty routine, which is you start with the you consider the, um, warm of the equation. What does that tell you about the solution? In this case is a method of under suspicions. And from there, once you approached the on are the method of undetermined coefficients. You just saw that. So you look at the homogeneous equation from the homeless hopes or the homogeneous equation. You go to the nine homogeneous equation you set down on. You find Matt, and then the Nano Virginia's equation formed and leisure to an IVP, and you do the eye of ups. So it's sort of it's again, just a matter of steps and going through all the steps correctly. Okay. And we'll see a few more of these pretty shortly anyhow, thank you very much.

Good day. Sorry about that. Good day. Kind viewers. How are you today? Um, so we're looking at problem number free here. Um, we're gonna try and solve this, um, problem and a couple of notes before we get started. Um, this is this equation is a second order. Um Why, uh, the difference, Your equation, the highest order or the highest power of the differential equation is to is linear. And it is constant coefficient. In other words, the coefficients of the function. Why are all constance and not variables a function of time? So this kind of tells us the method we use solving, um, this equation and it's ah, uh, in these cases, it's very It's pretty straight forward. Um, but it it's good practice in a number of times. So how do we go about doing that? Well, first off, we just assume y calls give a r t, um, is a solution, and we find the auxiliary equation. And by this, what we do is so if you plug y ankles are t to the r t, you're going to get of this expression again. So you very straightforward. And, um, you know that then factoring out beat of Artie. From here, we get this equals to zero because we're assuming it's a solution. Now, of course, Um and this is important here, since Ito Artie itself, um, is never zero on we we end up with just this expression. Unequal zero. So, um, the key here is that this this part here, So So this is what we call the auxiliary equation. And on the auxiliary equation, you know, is what will turn up many times. So it's important. Oh, you know, to know what it is and you know, in this case is straightforward. It's a polynomial, and you'll notice that to solve, um, this equation because of this right here, um, we have to solve the auxiliary equation. You know, we have to solve a polynomial. Well, um, and I hope all of you know how to So, Paulo, no meals. That's not something I'm gonna show you. Who's I hope you know it. Um, but when you do it, you get, um, you get our equals negative, too. Ah, and our equals negative three. And that tells us then that the general form for the solution, um, should have This looks like this guy here. Okay, Now again, One point he made is that since this is a second since our differential equation was second order, we should have to, um, linearly independent solutions in the general form. So, um, and you can shack that this is true. The Vignali independent solutions are C one you need to the negative to tea and see to your negative three cheese. The C one and C two are just numbers. Second, it could be any number. Um, and the these these air. So these this is in fact, our general form or assuming that both parts of these are solutions In other words, to see one in a negative to t and C two negative three t are solutions to the original difference equation. This would be the general form. And so that's really what we have to do next, which is we have to check. So I want to just check the parts here. So I'm just gonna soon, for instance, that y equals b negative duty. Um, and then when I plug that into the ordinary, different, sir equation here, um, again, through just factoring out the even the negative to t I end up with this here, Um, and it's it's pretty straightforward to see that for miles 10 plus six is a fact equals zero. So that tells us then, that you know, the negative to T is, in fact, part of the solution. And you want to do the same things through for, um Well, I guess See, two times e to the negative to team. I didn't really, You know, I didn't do the c one time that, but it's pretty clear what happens. Um, but anyhow, so this tells us then that, um are why Ian again to t is, um uh, eight solution. And by doing the same thing with the the negative three t um, you'll find the same idea, and and so that's sort of the key to this problem, which is to solve the auxiliary equation. Um, And then that shock. So get the auxiliary equation and checked, and the checking part is important. Also,

So we want to determine the 1st 390 germs in the tailor Serious polynomial approximation for the given initial well, problem hand. The problem is second Jewish for why last? Why it goes to zero with the following initial conditions. Why was your oh equal zero and why prime 00 equals one. So the polynomial approximation if the solution with cans of falling form some eye from zero to end eyes derivative off Why add zero the wind in my Iife Factorial times export I So why it zero equals to zero? We already know that. Why promise hero equals to one than from the audience health and get second Jewish off. Why ask minus y So why double prime at zero iss minus y and zero So zero We can find the surgery which if by differentiating second derivative there's minus y prime surgery or different zero will be minus first year different zero it is minus one. We differentiate the first religion together Force one it becomes minus second Jewish for fly so short derivative of why add zero equals zero Because secondary give at zero equals zero and finally we compute feels derivative It is minus surgery Mitchell Foy and zero and gives us minus minus one or just one. So getting it all together we get why facts approximately equal to X minus X Cube or three factorial plus export five or five pictorial, which it goes to acts minus X cube or six plus export five or 120.