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Use the improved Euler'$ method to obtain four-decimal approximation of the indicated value_ First use use 0.05_0.1 and thenY' = 2x - Jy + 3, Y(1) = 6; Y(...

Question

Use the improved Euler'$ method to obtain four-decimal approximation of the indicated value_ First use use 0.05_0.1 and thenY' = 2x - Jy + 3, Y(1) = 6; Y(1.5)h = 0.1 Y(1.5) ~ h = 0.05 Y( 1.5) ~

Use the improved Euler'$ method to obtain four-decimal approximation of the indicated value_ First use use 0.05_ 0.1 and then Y' = 2x - Jy + 3, Y(1) = 6; Y(1.5) h = 0.1 Y(1.5) ~ h = 0.05 Y( 1.5) ~



Answers

Use Euler's method to approximate the indicated function value to 3 decimal places, using $h=0.1 .$ Next, solve the differential equation and find the indicated function value to 3 decimal places. Compare the result with the approximation.
$$\frac{d y}{d x}=x^{2} y ; \quad y(0)=1 ; \quad \text { find } y(0.6)$$

In lumber. Ah, for we have this y zero zero a cheap. Was there a point one? And do I already x's X plus? Why square? So we put extra paralyze their hair and ah, deal. I already exceeds a five plus beef. I've replied before, just just be five square, which is why square is one zero. So a five, it's Xterra. So this column is Ah g oath, Xterra y zero. And our new Why is going to be remember, It's a full mother. Wife Zira. What? Why? I pass. Why was why I So why? Why was why? Zero plus do RD X testy. This column which is G zero g off Xterra line zero. And the mood cli h terrified one. That's our Why, Juan. And you put what I want here, which is d five, and you just drag it. And because the former out mother and you got the answer, which we need. Why off their point six? So this one three point lie six. Where flies his fart rounded to two one point one two. Okay, what is becomes too? Okay. It was from this. All right? I don't know what happened here. but, uh okay, this is under there. Are you okay? I go. There was something wrong. A lot of it is the same symbol. Say my this card. Finland not this part is not correct. Those are correct. So the answer is going to be there. A point two one three seven. Their point two on four by the exile spread shirt There point to our four for lumber.

Either This problem. We were asked to use oilers method to find, uh, on approximate new miracle solution to a differential equation. So here the ingredients will need d y de exes they give us. This will be our function G as the textbook shows us in this section. So in this case, her function g of X and Y is just one plus Why? It only depends on why we also know that why zero equals two. That's our initial condition. Age, as for all of these early problems, is 0.1, and we're supposed to find an approximation for why a 0.6? All right, that's the information we have. Let's make a little table. That'll make Oilers method easier to follow so that each step, we'll write a current X value. We'll write our current. Why value? You will have to compute g of our current X and y Valium after we do that, if we think of all there's method, we will need to multiply by H, which is 0.1, and we'll take that result added to our current why value and that will give us our next why value and then Well, the start all over again until we get X to get all the way up to 0.6 and we'll be done. So let's use our initial condition. Since why of zero equals two. That means that on X zero starts at zero and Y zero is too. All right. Um, the G value is easy here. One plus why So one plus two is just three. That this step will right What we're doing multiplied by 0.1, cause that's R H value. So we get 0.3. Add that to our current Why value, which is too. So plus 3.3 plus two gives us 2.3. And so that is our next My value. Our next X value, of course, just bumps up by H, which is 0.1. And in fact, we can right in all of the X values, because we just keep increasing it by 0.1 each time it will be done. By the time we were at 0.6. Okay, so we have a new X and a new Why will feed those into G. So one plus why one plus 2.3 is 3.3 times 0.1 gives his 0.33 Add that to our current y value and he should get 2.63 two steps done. Take 2.63 That is our new. Why value? I knew the whole thing again. So one plus 2.63 is 3.63 times 0.10 point 363 is the answer there. Add that to our current y value and we should get 2.993 All right, so that's our new high value. Well, this copy it 2.993 Add that toe one like it just our new G value 3.993 multiplied by 0.1. That gives us a 0.3993 and that her current. Why value? And we should be getting 3.39 to 3. All right, we're getting there. We have a new high value of 3.3 19 3 Figure out G. We add one to that good 4.39 to 3 times 0.1 gives us point for 392 and then add that to our current y value will give US 3.8315 All right, this is the last step. Will have to do. So we'll add one to our by value to give us 4.8315 times times 0.1 gives us point for a three to add that to our current. Why value for 30.3147 Well, copy that down here, and we are actually done since our X value is now 0.6. So we're going around 23 decimal places and 4.315 That is our answer. And so that is as approximately equal to why 0.6 and we're done. Hopefully, this helps.

Not sixteen. Devour DX is two acts over. Why? Okay, that's there. You see all this massive exile Terrapin Awan. Why want equals now we've got a wide zero. So are you. You good free And our deal Our D axes to an axe over life. Your mother's era. So we do go. Yes, it's a lie is three d y over dx Nice. No longer for murder by five plus three is going to be to multiply five. You're wanted, Brian, If I to move, I've got my bebop and are knew I was going to be B five plus c five, but apply to your point. War. Yes, that's correct. Two five to motive Lie a surks due out about basics. Okay. Yes, yes, that's correct. Hey. Yeah, What we want here, Terra point of six. Okay, We need a lot more code. Um, great. It's the valiant three point wants for his sober three point wants for It's over. Toilets, Messer. Toilets. Messer. Three point one three seven. And now let's just calculate thie answer value. So d y o ver t x because two ax over. Why we come with a cloud. Why? Both sides and What if that DEA suffered a terrible remember And to get to the roof half wife square equals X square. Percy Initial Value Y zero three. So one ofthe most blustery square was here plus c See, it was a night or two. His half y square is worth X square plus nigh over to we want to know while point six. So one half y squared equals zero point six square plus nigh over too. The wind was This is three over five square ni over twenty five plus nine over, too. It's going to be eighteen over fifty plus nine hundred twenty five. Just Carl. Transfer all of those two fractions. So the Togo is two hundred forty three over fifty. Does this part, people? Is that and we multiply. Two, two, forty three over twenty five. Square it. Yeah. How do you think that's where I would go through? Forty three. Do you have about five? Siriporn wass seven six. The story point Want one eight. There's a to value in the difference. That's why, uh prime. So difference Live Prime Wilder's. Why? Because zero point zero one Nice. So knotted that beef. Okay,


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