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Case Western Reserve University

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Problem 32

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).

Answer

The solution to the initial value problem $\frac{\mathrm{d} y}{\mathrm{d} x}=y^{2}-3 y+2, y(0)=1.5$ was found to be $y=\frac{2+e^{x}}{1+e^{x}}$

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## Discussion

## Video Transcript

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.