5

[Apts] Problem #I) Given y' (t) = Zty? with _ Y(to =0) =10) By hand calculation. solve the diflerential equation using the Euler' $ method . Use 0.2 and d...

Question

[Apts] Problem #I) Given y' (t) = Zty? with _ Y(to =0) =10) By hand calculation. solve the diflerential equation using the Euler' $ method . Use 0.2 and do two slcps t0 tind 91 andyz 4t / = 0.2 and 0.4. respectively: 6) Repeat pAnt 4) with using 0.1and doing four steps: c) If the exact solution Is y(t) (1777} compare Yuur = "pproximations Irom part 4) and 6) with the exact solution , (0.4) Comment On the nesult;

[Apts] Problem #I) Given y' (t) = Zty? with _ Y(to =0) =1 0) By hand calculation. solve the diflerential equation using the Euler' $ method . Use 0.2 and do two slcps t0 tind 91 andyz 4t / = 0.2 and 0.4. respectively: 6) Repeat pAnt 4) with using 0.1and doing four steps: c) If the exact solution Is y(t) (1777} compare Yuur = "pproximations Irom part 4) and 6) with the exact solution , (0.4) Comment On the nesult;



Answers

Let $F(t, y)=t^{2}-y$ and let $y(t)$ be the solution of $\frac{d y}{d t}=F(t, y)$ satisfying $y(2)=3 .$ Let $h=0.1$ be the time step in Euler's Method, and set $y_{0}=y(2)=3 .$ $\begin{array}{l}{\text { (a) Calculate } y_{1}=y_{0}+h F(2,3)} \\ {\text { (b) Calculate } y_{2}=y_{1}+h F\left(2.1, y_{1}\right)} \\ {\text { (c) Calculate } y_{3}=y_{2}+h F\left(2.2, y_{2}\right) \text { and continue computing } y_{4}, y_{5}} \\ {\text { and } y_{6} \text { . }} \\ {\text { (d) Find approximations to } y(2.2) \text { and } y(2.5) .}\end{array}$

Suppose we have a differential equation and we're trying to use linear approximation by Oilers method to find our next value at a given interval of time. So we would set this up like point slope form so we'd have Delta y, which is equal to y tu minus y one is equal to our slope. Times are t two minus t one which is Delta T and we'd move. Why one to the other side toe isolate white too. And we're going to condense this to Delta T. We're gonna add why one now from here? What we're going to do is will use our delta T that is provided by the problem are why one is going to be our initial condition and our slope is going to be our derivative. So to find her derivative now, keep in mind that this is dictated by both tea and why. So we'll have to plug in t and why t at 0.0, Wyatt Wyatt 04 that is equal to four are slope is going to be four. Delta T is 0.5, nor why one is four and this gives us two plus four, which is equal to six, which is our first. What's just our first approximation? For a second approximation, we do the same set up our why two will be different. Our Delta t would be different Are Delta t will double since we're going to the next interval. So instead, appoint fire, we will use one as Delta T Our M would stay the same because that point and we would add four and we will get eight as our second approximation.

So in this problem, we need to find out first to s steps for them. I learned method, so we call it that if the difference in the question is, why primarily? Questo 1/4 divine with a zero in some ways. Hero, then Okay, I learned. Mother says that u N plus one is u N plus f off D N Y in Delta t nowhere. This U zero is the initial condition. Why zero d m. So sorry. This is human. DE en is three in minus one, plus Delta T and yet 30. Is this tape sex? So in this problem, we are given with if off the way is deep less wife on dwhite zero is four The step size Delta T is your quantified and therefore, if you calculate u n plus one, which is u N plus a four p n u n delta t So that is u N plus tm plus u n and so one fight. Okay, Therefore, we have the relation u n plus one question you had plus t n plus human. I'm so I'm writing this 0.5. Yes. Okay, so let's compute step one state. So you won you will be you zero plus. So basically, t zero is the starting point. So t zero is zero. That's the thesis point. So this is basically t zero. So you zero plus de Jiro does you zero time sob, But this is nothing but Zero is four plus it's zero plus four is four games half. So this is six. So you running So So you want e questo six. The step to you Do a question for you one plus t one plus you one Diane's hub Now you one is already calculate is a six Stephen is t zero plus Delta t So that's so basically disease t zero plus 30 that 0.5 So 0.5 plus you wanted six himself. This is six plus 6.5 by two, which is nine point good fight and therefore you too. Easy question 9.25 Hence, regard the cool steps you want in question six and you to request 09

So the problem is Y. Prime is here with the three plus t minus Y. And we're getting a wide here, it was one. Um So for a yeah we're using T. Is equal to or this is what we're using the whole time. The T values will be using for this oil is approximation is two years ago to 0.10.2 0.3 0 .4. So for part A.R. H. value is going to be 0.1. So if we begin here, our first step, I'll leave with step zero. Why not go to Yeah. Um One because it's given the condition is wives here, it was one. Right? So that's our first step. Step one would be why one is equal to um F of this is a formula described as one of the first time T not of why not times age plus why not? Which is the same thing as going three Plus 0 -1. Time 0.1 Plus one is to go to 1.2. Now we have Y two, it's going to be equal to three. Then we're moving on to our t. Of one. Who's going to 0.1. Do you want one -1.2. Time 0.1 plus 1.2. That is 1.39. My third step why three around three plus zero to minus 139 Times one. Earth. Right? Yes. Time 0.1 plus 1.39. And we get one 25 7. 1 Step four. What 4 three plus 0.3 minus 1.571 time 0.1 Plus 1.571. As many vehicles to one 7439 Step five by 50 to 3 Plus 0.4 minus 1.7439 times 0.1 plus 1.5 sorry 1.7439. And that's going to be equal to one .90951. And that is our approximation using h value 0.1. No we don't step B Uh using the value of age uh 0.5-0.05. So it against zero is the same where saying why not to be one? So 1 1 is equal to. So this is going to be three, then start t values remain the same. So this is um you zero minus one but now it's going to be times 0105 plus one. That is 1.1. Yeah two what? Two is 0 to 3 plus 0.1 minus 11 Time 0.05 plus 1.1. That is going to be 1.2 three 53 is equal to three. Close zero to minus 12, 0.05 Plus 1.213. When I probably see the pattern, Why four is three plus 0.3 minus 1.3 times 0.5 plus 1.3 21.4. So it lasted here is five 550 to 3 plus 0.4 piracy 0.4 miles 1.4 times 0.05 plus 1.4. That's 1.5. And that is our approximation for H values here. 105 Part C. We're using H. b 0.025. So same steps. Wh why not feel too warm one is why one is able to three plus zero minus one times 0.25 plus one. Which is equal to 1.05. Yeah scroll down. Remember this to Y. To the Z. Three plus 0.1. Mya's 1.0 five times 010 to five plus 1.5. And that is 1.101 253 of Y 30-3 plus zero point to my s 1.10125 Times 0.25 plus 1.101. Yeah. 1125. That is all equal to 1.15371875. Yeah. Step 44 Is equal to 3:03 -1.153 71875. Stop reading because I'm getting really long. But this is equation mhm 5371875 is able to 120737 5781. I said one more to go. Yeah. Mhm mm. My answer. Previous value. Yeah. Yeah mm. Times are H. Plus the previous value. Yeah that this is comes out to be 1.26-191386. Which is our approximation using h. value of 0.025 the party to solve. And as you can see as our H gets smaller are approximation also gets smaller as I assume or we can assume more accurate to the true value. Mhm. Alright so here we're gonna solve and evaluate each point. So we're gonna have our equations again. Why prime plus I'm ruined the otherwise over the size plus Y. As you can see three plus T use our same formula. Mputu go one F. F. T. To enter L. D. T. Or one bt She was a. T. So a multiplying factor is E. To the T. Remember by that out I. E. T. Y. Prime plus E. To the T. Y. 63 E. To the T. Plus T. E. To the T. I don't make a product. Wellness. I was going to eat the T. Y. 53 E. T. Plus T. E. T. All right I'll integrate both. Have the respect T. So yeah. Yeah. Okay playful. I'm gonna separate out left the right hand side to make it easier. Now we'll get these will cancel how easy T. T. Y. It's gonna eat the T. 2 3 E. to the T. Plus E. To the t minus. Oh sorry plus T. E. Two T mice E. To the T. Plus C. So I'm gonna divide get the Y. Alone. Don't get wise equal to three plus t -1 Plus C. So why is you go to two plus T. Plus C. Is the nationalization of Y zero equal once around +12 plus zero plus cease to see negative one. The final conclusion is why equals one one plus T. That's not his need to evaluate this at each point. You compare so Why a 010211. Why a 0222 1.2 Y. 0.30- 1.3. And why is 04 is equal to 14? Which makes us think that our H. For number two is the most accurate um across the nation.

Suppose we have this differential equation and we're trying to set up in Oilers method. Linear approximation. So let's start with her formula. Why two is equal to m times ti tu minus t one also known as Delta T plus. Why one where? Why one is our initial condition. M is the slope better derivative? Uh oh. Are derivative at the point. Uh y equals zero are y equals negative one t equals zero and our delta T, which is 0.2. So our first approximation. Why, too, Let's go with you. One are approximation you want is equal to em times 0.2 or 0.2 time plus a negative one. And our slope at this point is going to be negative one times negative one which is equal to one. So our slope one So why two is equal to 0.2 minus one which is equal to negative 0.8 which is our approximation one for our approximation to the same thing except our double or delta T is doubled. So new delta T is equal 2.4 instead of point to So our second approximation same slope of one 0.4 plus a negative one. It's going to be equal to negative 0.6. You too


Similar Solved Questions

5 answers
What color Thc the I genotype frequencies H for 4 in the population? the allele for the color green populution Calculate - snakes V to at least two decimal places: 50 turquoise, Bohd alleles demonstrate grecn: 1 color blueallele frequency 8 What is the allele frequency of B" this population? L places.
What color Thc the I genotype frequencies H for 4 in the population? the allele for the color green populution Calculate - snakes V to at least two decimal places: 50 turquoise, Bohd alleles demonstrate grecn: 1 color blue allele frequency 8 What is the allele frequency of B" this population? L...
5 answers
Given:AR(q) =24q + 642 9"For what number of q is the maximum possible total revenue avilable?
Given: AR(q) = 24q + 642 9" For what number of q is the maximum possible total revenue avilable?...
5 answers
Glu 33 nM + Glu 33 nM siRNA ET-1 siRNA ControlL 1 1 1 1 V ; 1 1 L 1 1 U ? 1 0 1 E L 318
Glu 33 nM + Glu 33 nM siRNA ET-1 siRNA Control L 1 1 1 1 V ; 1 1 L 1 1 U ? 1 0 1 E L 3 1 8...
4 answers
3.2 (Matrix space and annihilating polynomials) Determine the dimension of Kmxn . b) Let T € N and p € Px; P(x) Ci_o " aixi and interpret it as mapping p : Knxn 5 Knxn viap(A) : C aiAi (Ae Knxn) i=0while using the convention AO = (identity matrix)_ Show that for every A e Knxn there exists p € Px {0} such that p(A) = 0. (2+4 P.)
3.2 (Matrix space and annihilating polynomials) Determine the dimension of Kmxn . b) Let T € N and p € Px; P(x) Ci_o " aixi and interpret it as mapping p : Knxn 5 Knxn via p(A) : C aiAi (Ae Knxn) i=0 while using the convention AO = (identity matrix)_ Show that for every A e Knxn th...
5 answers
Detenine whether or not W is where W consists subspace of RI of all vectors (a,b,c) inR" such that: (a) b _ 02 yes nO (6) ( = 2 = Jc es nO (c) 36 yes nO (d) (b = 0 yes nO:(e)u < b < €yesnO(D)"+b+0=0yesmo
Detenine whether or not W is where W consists subspace of RI of all vectors (a,b,c) inR" such that: (a) b _ 02 yes nO (6) ( = 2 = Jc es nO (c) 36 yes nO (d) (b = 0 yes nO: (e) u < b < € yes nO (D) "+b+0=0 yes mo...
5 answers
Your reasoning: 0 9'(~2)9'(0)9'(2)9'(4)y=g(x)
your reasoning: 0 9'(~2) 9'(0) 9'(2) 9'(4) y=g(x)...
4 answers
The following reaction has been described in the chemical literature and proceeds in good yield: This reaction may be more complicated than those you have s0 far encountered: Nevertheless, on the basis of what you have already learned; YOu should be able to predict the principal product: Draw the principal product for the following reaction: Click the "draw structure" button to Iaunch the drawing utility:Nadraw structure
The following reaction has been described in the chemical literature and proceeds in good yield: This reaction may be more complicated than those you have s0 far encountered: Nevertheless, on the basis of what you have already learned; YOu should be able to predict the principal product: Draw the pr...
1 answers
Evaluate the integral by changing to polar coordinates. $\iint_{R} x y d A$, where $R$ is the region in the first quadrant bounded by the circle $x^{2}+y^{2}=4$ and the lines $x=0$ and $x=y$
Evaluate the integral by changing to polar coordinates. $\iint_{R} x y d A$, where $R$ is the region in the first quadrant bounded by the circle $x^{2}+y^{2}=4$ and the lines $x=0$ and $x=y$...
5 answers
Given 'k)-r_ 42 (a) Find f (r + h) and simpllfy: (b) Find {(rth)-Ll) and simpllfy:Part: 0 / 2Part 1 of 2(a) f (+h) =O+0 0-0
Given 'k)-r_ 42 (a) Find f (r + h) and simpllfy: (b) Find {(rth)-Ll) and simpllfy: Part: 0 / 2 Part 1 of 2 (a) f (+h) = O+0 0-0...
5 answers
18. Evaluate the expression exactly if possible Ifnot possible; state why tan tan
18. Evaluate the expression exactly if possible Ifnot possible; state why tan tan...
5 answers
Vector 1 is 41.0 in long and makes an angle of27.0 degrees counter-clockwise from the negative x-axis. What isthe x-component if vector 1 in inches?
Vector 1 is 41.0 in long and makes an angle of 27.0 degrees counter-clockwise from the negative x-axis. What is the x-component if vector 1 in inches?...
5 answers
Find all the second-order partial derivatives of the following function: w = 8x sin (7xy)4 1 J w dyor J Ux
Find all the second-order partial derivatives of the following function: w = 8x sin (7xy) 4 1 J w dyor J Ux...
5 answers
Question 18 ptsA total cost function; in thousands of dollars, is given by C (a) 6q? 254 where the quantity q is measured in thousands and 0 < 9 < & At which production level is the average cost minimized?Clearly state what is your average cost function and show all the relevant work: Do not forget to explain why you have a minimum!
Question 1 8 pts A total cost function; in thousands of dollars, is given by C (a) 6q? 254 where the quantity q is measured in thousands and 0 < 9 < & At which production level is the average cost minimized? Clearly state what is your average cost function and show all the relevant work: D...
5 answers
You have dissolved 19.751g of barium chloride in water: If all of the Gibb's Free Energy liberated by this reaction is transferred to a 44.303g block of copper at 20.439C, what is the final temperature of the copper block? The specific heat of copper is 0.3845' g*'CIsubstancelAG?t moll BaClz(s) 806.67 Ba 2aq) 560.84 Cl- I(aq) 132.78{Assume 100% energy-to-heat transfer efficiency for this problem:}
You have dissolved 19.751g of barium chloride in water: If all of the Gibb's Free Energy liberated by this reaction is transferred to a 44.303g block of copper at 20.439C, what is the final temperature of the copper block? The specific heat of copper is 0.3845' g*'C IsubstancelAG?t mo...
5 answers
Show enough work to make your method of solution clear Consider the region in quadrant oe bounded by the curve y = Vzx the Line y = x - 4, and the axis. Sketch the region in quadrant one showing points of intersection and "revolving" approximating triangles you will beFind the volume of the solid generated by rotating the given region around the line -3
Show enough work to make your method of solution clear Consider the region in quadrant oe bounded by the curve y = Vzx the Line y = x - 4, and the axis. Sketch the region in quadrant one showing points of intersection and "revolving" approximating triangles you will be Find the volume of t...

-- 0.019583--