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Points) Consider the iuitial value problemV)6)(z _ 6)(1 _ 6)z6 = (2)f (8)f |For which of the following values of %o does there exist some t4 > 3 such that u(t4) ...

Question

Points) Consider the iuitial value problemV)6)(z _ 6)(1 _ 6)z6 = (2)f (8)f |For which of the following values of %o does there exist some t4 > 3 such that u(t4) = 2yo? s0- Jo = 1 Yo = L5 Vu = 2.5 3.5(VII)points) For which of the following initial value problems docs the solution exist for all + > (? v(t) '(#)f 3(o) B. y () 2(16 = (0)f y(t)3 (0)6 V (6) v(')2e-y(t) 4(o)

points) Consider the iuitial value problem V) 6)(z _ 6)(1 _ 6)z6 = (2)f (8)f | For which of the following values of %o does there exist some t4 > 3 such that u(t4) = 2yo? s0- Jo = 1 Yo = L5 Vu = 2.5 3.5 (VII) points) For which of the following initial value problems docs the solution exist for all + > (? v(t) '(#)f 3(o) B. y () 2(16 = (0)f y(t)3 (0)6 V (6) v(')2e-y(t) 4(o)



Answers

Consider the initial value problem $y^{\prime}=y^{1 / 3}, y(0)=0$ from Example 3 in the text. (a) Is there a solution that passes through the point $(1,1) ?$ If so, find it. (b) Is there a solution that passes through the point $(2,1)$ ? If so, find it. (c) Consider all possible solutions of the given initial value problem. Determine the set of values that these solutions have at $t=2$

Hello and welcome to problem 21 of chapter two, section four, you're asked to consider the initial value problem. Why? Prime? Nichols wider the one third. And were asked if there is a solution that passes through the .11. So find it. We also asked if there is a solution that passes through the point to one, find it. And we'll talk about all the possible solutions on the uh given initial value problem. So let's start off with uh talking about there are 2.4.2 in this. It says that if the if why where it's a partial derivatives with respect to y is uh has a discontinuity um then there is not guaranteed a unique solution in that region. So why the one third doesn't have any discontinuities but its derivative does. So if you let f be equal to white prime, then partial F partial lie is able to one third wide of the negative two thirds which has a uh discontinuity at y is equal to zero, which also happens to be our initial value point. So what does this mean? It means although a solution exist it All right. Although a solution exists, it will not be unique or it may not be unique regardless. Let's solve this. So um we integrate both sides will be left with um old intervals do I? It's equal to that you know of uh Sorry, let's let's restart. We'll do by prime or do I. D. Too Equals wives of the 1 3rd from that. We'll have dy over y two the one third because DT and here we can just integrate we'll get two thirds why to the 2/3 is equal to T. Plus C. And then we can uh you can solve this for why we get see have a I believe this will be 3/2. So if you sell this for a while that y. Is equal to um 2/3 t plus c. And All to the 3/2 power. Remember this is also a plus or minus because there's a square root there. Uh huh. All right. So from here we can plug an initial value 00 just looking at it. Uh C is going to be equal to zero so we can cross that out. Um simplify this a little bit 3/2 and let's see if one is a valuable valid solution to this um differential equation. So we plug in one here we get to the sun side so one is equal to plus or minus uh Two thirds times one. Well that's uh that's two thirds 2/3 to the 3/2. Um Well we can cubits term take the square so also one is equal to Plus or -8/27 square root but this is not valid. So the answer is no. What about for 2:01? Um some argument one is equal to plus or minus Uh 2/3 times two. The three halves. Um And just for inspection no this is not in the solution. Um What about the next part? Consider all possible solutions with the given its value problem. In terms of the set of values of these solutions have its equals two. All right, so let's plug into equals to them. So why is equal to plus or minus 2/3 times 2? So four thirds To the three house? Um So why is equal to closer minus? Well, four cubed is 64 square, that is eight. Um Three cubed is 27 square, that is square to 27. But these are the solutions that T equals two and that includes the problem.

In the question. We have to find militarization or the given function f X almost might, which is a little suspect this guy spend less exactly spread at that point, one was even see back, not moving what is, in fact, one month or 100 exactly for people to access for Let's buy scrapped 70 square Act one on one. Similarly, F y. Is that one of the organization for the event from, for a part, one last one text minus one plus zero line minus one. What's be effect on a month on my feet effects that, except in a quiet way. So the immunization for being won by just one by no foxy effects at why that before the sea and the thank you.

In the question. We have to find billionaire ization 40 function. And if it's why is that which is exploit blessed? Why isn't less explicit at the given point? One on one on one B one on my several moments ago See Park zero now words solution for a given approach like we were to treat effects. One. Events that external white, my white bless Zak So effects that one Really f y i x from life heads. Why? Why bless? That's at one form of one. So the unionization for the given function will be free. Blessed to X minus one place to why minus one. Bless exact My nest, which wet? That's why bless. Who's that? My nest? No moving. What's it like to be if there one So did unionization are to get one big? Were consider less evil. It's my next one. Less by my guest by minus C, less set minus C, which will be quiet. Blessed not going to work. In fact, wait for Cruz. Even effects likely. Why? And f 000 So the organization for the sea for a given function We see this with the final answer for the question. Thank you

In the question. We have to find the immunizations for the given on effects on the white. Understand, which is a close to excess Where less vice well plus z squared at the point, eh? About +11 form of be part still move on. Steve went on, moving forwards dissolution or give us a function on one on one with three at exactly 11 moment. One which would tie it to buy that for the given function, but a less too x minus home less to buy my next one less to finance. It will make us less wine, less tools. My next three now moving works of art. So the unionization for they will be one plus zero x minus. Really blessed to why minus one less zero minus which by my next one, see one way effects that one and that one for a given for parts B wets minus one. And this one final question. Thank you


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