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Problem #7: Considcr thc diffcrcntial cquation [15 marks] 4xy" + (2+x)y'-y = 0.Without solving this diffcrential cquation, onc can say that any powcr scri...

Question

Problem #7: Considcr thc diffcrcntial cquation [15 marks] 4xy" + (2+x)y'-y = 0.Without solving this diffcrential cquation, onc can say that any powcr scrics solution centcrcd at thc ordinary point xo 5 (i.c_ scrics involving powers of x 5) has radius of convcrgcncc at lcast cqual to R. What is R? (Entcr infinity if R ~,)Find the two roots of the indicial equation associated with the regular singular point Xo 0. Scparatc your answcrs with comma_ (c) If r1 denotes thc largcst of the 2 ro

Problem #7: Considcr thc diffcrcntial cquation [15 marks] 4xy" + (2+x)y'-y = 0. Without solving this diffcrential cquation, onc can say that any powcr scrics solution centcrcd at thc ordinary point xo 5 (i.c_ scrics involving powers of x 5) has radius of convcrgcncc at lcast cqual to R. What is R? (Entcr infinity if R ~,) Find the two roots of the indicial equation associated with the regular singular point Xo 0. Scparatc your answcrs with comma_ (c) If r1 denotes thc largcst of the 2 roots of thc indicial cquation, the diffcrential cquation above has solution of thc form Cnxn+n n=0 Whcrc g(n)cn-1, n 2 1, for certain function g(n). Compute g(n). Writc thc sum of thc first non-zCrO tcrms of the scrics in (c) if c0 = 1. What is the radius of convcrgence of the scrics in (c) if c0 12 (Enter infinity ifit is infinite.) Enter your answer symbolically, as these exampes Problem #7(a): Enter your answer symbolically, these examples Problem /7(b): Enter your answer a5 symbolic function of n, as in these examples Problem #7(c):



Answers

Analyze the solution $ y=\phi(x) $ to the initial value problem
$$ \frac{d y}{d x}=y^{2}-3 y+2, \quad y(0)=1.5 $$
using approximation methods and then compare with its exact form as follows.
(a) Sketch the direction field of the differential equation and use it to guess the value of $ \lim _{x \rightarrow \infty} \phi(x) $
(b) Use Euler's method with a step size of 0.1 to find an approximation of $ \phi(1) $.
(c) Find a formula for $ \phi(x) $ and graph $ \phi(x) $ on the direction field from part (a).
(d) What is the exact value of $ \phi(1) ? $ Compare with your approximation in part (b).
(e) Using the exact solution obtained in part (c), determine $ \lim _{x \rightarrow \infty} \phi(x) $ and compare with your guess in part (a).

All right. We are observing a population with mean 4.8 and we want to calculate the sample mean and standard deviation for the following sample data obtain to do this. We we know the definition of expire and S for a sample which are given here, X is equal to some of the data divided by n or 4.4, and s is equal to the sum of the deviations about the mean square, divided by n minus one or 0.28 Next we want to implement a left tailed test on this particular population, so we want to test the population is actually less than 4.8. Using a significance level output equals 0.5 Where we are told that X is approximately normal, that is this distribution is approximately normal. So we have to answer the following questions. In order to complete this test. To start off with, what are the significance of hypotheses? Alpha equals 0.5 No hypothesis H is new at this 0.4 point eight and H. A. Is that me was less than 4.8. It's the left tail test. What distribution When we used to be the test statistic, we're going to use a student's T distribution because sigma is unknown. We know that we can do so because the shape is symmetrical and bell curved. Given this, we want to calculate the T stat which is given by this formula. The T stat here equates a negative 3.499 Now, given this T stat, we want to compute the P P interval and then sketch the associated students distribution for P. Since we have degree of freedom, uh six minus one equals five. We have the R. P interval is between 50.5 point 01 We can grasp this as the area to the left of our T stat highlighted here in yellow and Marcus P. From this, we can conclude since P is less than legal to alpha, we have statistically significant findings and we can reject R. H. Saw, which we can interpret to ultimately mean that we have sufficient evidence suggesting our population means is less than no means for pointing.

Okay, So for party, we have that. Why of zero equals two implies that C one is equal to And why have pie over to equal zero implies that C two is equal to zero. So then we can write why of X is equal to co sign of X And this is our unique solution our unique solution. So this is part of a no part B. When we use the same kind of approach, we will get that Why was your equals? To imply I see one equals two again. But why have pi equals zero implies that C two is equal to Well, nothing because actually, the equation will vanish. So we'll get zero equals too coarse and pie plus c two sign pi So see to sign pi is C two times zero which is zero to see too vanishes and you got zero equals negative too, which is, um not true. So we have no solution for this part. And for part c, we have C one equals two and we will obtain see two equals two is well, actually sorry, I take that back. We don't get seat equals two. We get negative two equals negative too. And this is coming from the initial condition. Why have pie is equal to negative too? But this is true no matter what. So there are infinitely many solutions for part C infinitely many solutions.

Hope close que for pretty early, so quiet Make my no make e make a thing We make a square My name is E O You're e make us queer times e Yeah, he tames okay, make it. Oh, my All right. Oh, man. Cheese Solve the equations you couldn't abuse are line is produces a he told u K and five k bringing in and b people he will be Oh, make us so I got dio e and time because weird more school year quick and make a square um, following you for using graphs Equations be quick one in math so pleased upon change in u A e replace it with a vote here looking to buy the pain is only square squared here one next make a multiparty divided by 25 points Omega square squared. I hear your point, Omega Square thing. You gotta pluck the craft. So here it is, your hero. You have a mega here on the x axis and the other one is either a RFI. So at five, where me equal spy, It's gonna be undefined through its from his youthful life site. Good to the left hook. Boom zero But as you were close to closing five keep going upward. I was thinking. But if you go Chris, you go from five always looking. It is going from that YouTube video all the way to zero. That's just for just these two. Here they're upset. But along this axis be usual. It's continents for two of the girls Goes from envying be I see for you see, it'll be using be for Vera and equaling one. And he going 25. You replace it with the age for one time. I mean, it's wrong because clear, be here. Absolutely a line Kirk going from veal at That's where actually his world. Sorry, who make I pulled off? Oh, hey, is course five. But hashing goes further That my gruesome thing because zero that this Europe since a Thea other one second line it's gonna be zero. So this is dispersant he in this represents you can t yeah equals zero and make inkling equal square Rube a him then a wasn't being and the leak was here and then he's two times I asked him to make a English welcome looking T sign Homemaker used y prime two things in past times Don't make inverse multiplied by Make it he plus or making tea I don't time to tens baskets but maker and the injured we'll talk to me Might is square Keep staring, Nick between I made a decent schools here are here so get the vast times to make and inverters to may as well make it squared times getting time May I was a kid his two times and his Ramaker is the key for a using a making equal squared Ok, I will make hand me not secret.

So it's also sketch did actual field for a differential equation. Knobs off, sketched out, and I'm gonna show you in a moment What I got now This is what I got's. And then we're told to guess the value off limits as X test in finishing. Now we have. That's why 00 you called to 1.5. So this is one. So let's say this is wild zero. So we have that the limits as X test infinity or five eggs. You can make a guess. That's it's probably going to be two or one because it's in between one and two. So let's make a guess. That isn't also one, Andi confirmed. That's in question number eight. By making a guest, that's gonna be one. You can make a guess off to also. Yeah, that doesn't Monsour. Now let's look at question number B. Now question will be once us. So use the last method with step size age record zero points one to find the approximation. Five off one. No, recalled Ula, met owed. Now we have. That's why of zero you called a 1.5. This implies that x zero quartz zero and then y zero quartz a 1.5 Now our F off extra mile. Why records y squared minus three. Why plus two this implies. And then by you as men Toad we have. That's why I n plus won't record. So why in blows age half off extend comma y end which in this case, will be why end plus 1/4 white and plus my h zero points one and then my f off x and y and will not be why and squared minus three y n last soon So this is all gonna juice So we know why zero already So to find why won't you take end to be 1/4 0? Now I make I've made it simple and then off calculated these So let's make it simple and explain on doing why end? So when n is one, we have zero points one which is 1.4750 and then when any Sue we have zero point soup which is 1.4501 when and his three. That's excellent. Does up on three of, um 1.45 to 3. You can continue these all along and then we get to end. That's X. And what end? We get any quarter AIDS, you have zero point AIDS and that's one point threes or it's four and they have nine. You have 0.9. That's one point to it. Several one and then saying which is 1.0, which is what we're looking for. Its one point so 666 so you can fill in the the samples. You're just using this formula. This is the formula you're using now. This implies that our fire off alone is approximately 1.2666 now for the seed pods. We're told to find the formula for five X. No, we have. That's Dwight. The eggs he called. So Y squared minus three. Why close to divide was said by Y squared. You have D y over y squared minus Terry. Why? Blast tube records DX into grids, both sides because it's separable. So what's the integral of these? We have to use special function now recall. That's or we can observe that these is just equal to why minus one. Moz played by wide lust soon, so why do for us this solved the partial function of these. Why my nose swan off y plus two, that would be a over. Why minus one plus being over white lost soon. When you're more supplied by this denominator, you off one because so a off y plus two plus B off y minus one. No, let's why he quotes one. You have that you're a because of minus one. Let's why quarter sue you have that you're being is important one. So this implies dance one over. Why minus won't Why minus to be called So minus one, which is my a o ver y minus one plus being over. Why minus two said at one place. That's when I go back to distribution star. I can write it as so stop becomes into girl off. All right, this has won over. Why? Minus two minus one over. Why minus one, Then do Why? Because this is the partial fraction of solved. Recalled to into girl off DX, way into good bees. You have natural love of absolute butt off Y minus two minus natural off. Why minus one equal to eggs plus c, I can rewrite this as not your love off. Why minus two over. Why? Minus one. You want to explore? See now, before I go on, I'm gonna get the initial value off. Why? Off zero it towards 1.5. This implies duds. Not sure. Log off minor. Zero points. Five over 0.5. Recall two X zero lost. See, this is equivalent of natural look of absolute ball off minus warn equal to see which is not your local one, which is equal to zero, not Charlotte of 10 Okay, I don't need an absolute value here because of taking the absolute value, so I don't need this again. So the natural laws of war is zero now, Missy is 1/4 0 So this implies that I have natural law off. Why? Minus two over why minus won't record two eggs plus C, which is zero explanation. What sides? You have y minus two over my minus warm. You call to exponential of X. This implies that's why minus two over why Miner sworn quarts of plus or minus two X now this point and it's making choice. Would it be in close for minors? Now? Let's quickly use the initial value y zero records of 1.5, So this would be 1.5 miles two b minor, 0.5 over 1.5 minutes without 0.5 equal to plus or minus E to the zero. So this is minus one week or two. So see, that joins us to be minus one. So that means we don't get the blows. It is actually miners. And that's what would Dr using the initial value. So this implies that why minus two over why minus one equal to minus me to the X. Because for this to people you have to use minors. No, a curse. Multiply both sides. I have why minus two Because you minus eat the eggs off. Why? Minus one That implies the Y minus two He called to minus Lied to the ex off. Why lost to be ex? When you pour this the guy you have, why off? One lost eating the eggs. You also eat the explosive soon and then why? Which is a good fight of eggs is it also eats and X plus two divided by E to the X plus one undocks is our answer Now we're told to we're told to on graph I on the direction fuel. So when I got off, I have already graft doubts and I'm gonna show you That's in the moments. Oh, which is this? So it is this black line. That is our question. If the answer we got in Patsy, that's is this line between one and two Now for the deep parts, we're told you chocolates. Five won't set out. He lost to us. It won over Italy, one plus one. That is approximately one point June 6. It's 94 We were told to compare these with the answer we got for you. As metal in part be so from parts be, we aren't Doug's fire off. One was approximately in court. One point sure six in six. So we see that the D for the D for. Bye Bobby. Less done. 0.3 Now for the e parts. We are told to get the exact solution off. Didn't seen time determine the linens as X test in finishing off five eggs, which has got to be eating explosive soon divided by each of the explosive one. Now you can And I used the fact that eat the eggs will dominate in time in the numerator and denominator. Or you can also use this idea because we are limited extent. Infinity divide within the merit challenge Nominates about eating ex So you have one blows two over easy eggs divided by one loss one over E t X Ah, specs. Destination it. See this Ghost zero. This goes to zero. There you will see that the answer would just be limits as X test infinity off alone, which is just because of one on that conforms with the guests. I aren't that the answer is one and that's corn farms are answered.


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