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Euentfsc 21-2X. finl dsldr hy implicit dilferntiatun esluule Uhc dcrivative Mt thie %ivco pvol; (5U 24.0y020 (-1,0 (2 Ii (24 0n. oli Fuek 0up X U= (23)Kan hypetblsOx KatalcCunetEunhrs 19-4 lud Iltc slopr e Ibc Lo: pratuur IIllt Eauh 4u Il"ecn IlnlMuch uAcneCnad 0'" Mtume 4126d0 " CnxtkirmOx Axoxlnmt (21

Euentfsc 21-2X. finl dsldr hy implicit dilferntiatun esluule Uhc dcrivative Mt thie %ivco pvol; (5U 24. 0y 020 (-1,0 (2 Ii (24 0n. oli Fuek 0up X U= (23) Kan hypetbls Ox Katalc Cunet Eunhrs 19-4 lud Iltc slopr e Ibc Lo: pratuur IIllt Eauh 4u Il"ecn Ilnl Much uAcne Cnad 0'" Mtume 4126 d0 " Cnxtkirm Ox Axoxl nmt (21



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In Exercises $19-26,$ use implicit differentiation to find $d y / d x$ and then $d^{2} y / d x^{2} .$
$$2 \sqrt{y}=x-y$$

We have y prime equals minus y squared plus 10, y minus 21. And want to get solutions of that differential equation for and why? Why attempt equals zero equals 1 and four. Now let's see here is yeah, three is 0 And I think seven is also a zero. Does that make sense? seven is -49 plus 70 brian. This trying to get So three and 7. So this has roots at three and 7. And it's very much like um the previous case, but now just just shifted. If we start out here, why is positive but the derivative and negative. So we're gonna go to smaller values of why and then smaller and smaller and then get basically smaller until we go off to negative infinity. Now, if we go to over into this region here, we say at four we have a positive derivatives that we're going to go to larger values of why and even larger values. And then we're gonna deliver is gonna start to slow down Uh and we're going to approach this point here at seven. And likewise, if we started over here, we would approach into this point here. So when we started one, we basically rapidly go off to minus infinity, I guess that would be here and we started four. You can see that for a while, I think the inflection point where this is the maximum that when y is five, so we have positive coverage er here because the slope is increasing and then the slope is decreasing Until it approaches zero. Um And as White Horse to seven, so we have something like this and again, quite a bit like the previous problem. But again, if we started up here, we would get something like we had over here.

Right. So our initial value problem is going to be the third derivative of why. Plus two. Why two times the second derivative, minus nine times E first derivative minus 18. Why is equal to negative 18 X squared, minus 18 eggs plus 22. So the corresponding auxiliary equation is our cubed plus two R squared minus nine. AR minus 18 is and we set that equal to zero. If we factor it, it's gonna be the same as AR minus three times R plus two times our first three being equal to zero. So that means that are ours are equal to three negative to negative three. So the homogeneous pushing of the solution is some constant times e to the negative three x plus some other constant times e to the negative two x plus some third constant times e to the power of three x So now we want to find the particulate a part of the solution and that will be in the form X squared plus B x plus c. So the first trip did of that is gonna be to a X plus B thesis. Aiken, derivative of our particular solution will be to a and the third derivative of the particular solution zero. I'm gonna put all these into this original. Ah, problem and we would end up with is zero plus for a minus 18. A X minus nine. Be minus 18. A X squared minus 18. Be eggs minus 18. See which is equivalent to was gonna put things. Ah, instead order. So a negative. 18 a x square plus negative 18 a minus 18 B, times x plus for a minus nine B minus 18. See? Okay. And so that we wanted to accept this equal to what was on the right hand side of that initial value problem? Yes. So this is going to be equal to negative 18 X squared, minus 18 X plus 22. Okay, well, so we want to compare the coefficients. So we have, um negative 18. A needs to equal negative. 18. So that would imply that a is equal to one. Came in for the second one. We have negative. 18 A. We know that a zygote a one. So the night of 18 times one minus 18 times be needs to equal negative 18. Okay, so that means negative 18 minus 18 b Mystical native 18. So this surprise that b is equal to zero their last one four times a just one minus nine times zero, which is zero um minus 18 c needs to equal 22. I guess this means native 18 c is equal to 18 so C is equal to negative one. So then we have, uh, the pieces we need for our particular solution. And that means that the particular solution is equal to is gonna be a X squared is one such as X squared plus B x p zero ah plus C minus one. So x squared minus one is the particular solution. And so if we add our particular solution to the Virginias portion, we get that the full solution is equal to x squared minus one plus some constant times e to the negative three x plus some other constant times e to the negative to eggs, plus some third constant times e the three x. Okay, so this is really our general solution and now we want to start using our initial conditions. Now, one thing to note is that the first derivative here is equal to two x minus three times. See one times e to the negative three x minus two times C two times E to the negative two x plus three times C three times either three X, and the second derivative is equal to two plus nine c one e to the negative three x plus four c two e to the negative two x plus nine C three. Either the three x It's not gonna put our initial conditions into these equations. For example, we get why zero, which is gonna be negative one plus C one plus C two plus C three is equal to negative, too. So that means that C one plus C two plus C three must be equal to negative one from the first derivative from why primal zero. We get negative three c one minus two c two plus three C three is equal to native eight and Leslie from the second riveted at zero. We get two plus nine c one plus four c two plus nine C three is equal to negative 12. At least try to from both sides and get that Mayan times C one plus four times C two plus nine times C three is equal to negative. 14 Your hands And then I saw this system of equations. So maybe I'll start by numbering them. Let's just say this one, two and three. So let's start by taking three times the first one and adding it to the second equation. And so the sea ones will get knocked out so we'll end up with See to Plus six. C three is equal to when we take three times negative one and added to the negative eight. So it's gonna equal negative 11. And then we can also take three times the second equation and add it to the third one. Yes, In that case, we're gonna end up with see one being eliminated again s and we have negative six c two plus four c two says the negative to see to it that we have a nine c three plus nine c three so close 18 c three asking equal the three times negative eat minus 14. It's going negative. 38 It is. So that's really leads to new ones as our equation. Four or five. So I want to take two times for equation for and and that to equation five. So the see twos will cancel out. It's gonna give us 30 times. C three is equal to negative 60. So I mean, C three is equal to negative too. But I think I'm just gonna go backwards and put that into another equation. So we can take C three is equal, negative two, And put that into the equation for so you would get C two plus six times Negative, too. So guests minus 12 is equal to negative 11. If we had 12 to both signs, we would get see you choose equal to one. And then we can put both of those into any of our first equations. First three that handle through my vote for C one plus C two plus C three is equal to negative one. So you would get C one plus c to just with one plus c three my eyes to people, the negative one on this means that C one is equal to zero so that we have the values for these constants. We can put that into our general solution we have here. Okay? And when we replace those Constance, our final answer is one of X is equal to X squared minus one plus e to the negative two x minus to E to three x and that will solve our initial value problem.

This question asked What type of function models the data. What we know is that in our White Column, if we multiply one times 1.2, we have 1.2. Then if we multiply the next two times the difference that we just figured out 1.2 times. 1.2 we have 1.44 Do it again, 1.44 times 1.2 we get 1.73 Therefore, if you're multiplying the same number as you can see, we're doing it by 1.2 each time. This is an exponential model.

Indestructible problem Number 22 since a It's that White Square minus truex, my menstrual. We have to find y rash on Dwight overnight there's deficient Bull states Villiger, Dior, DX off y squared minus uh, why square minus two is the constants that comes outstayed your body kicks off X will be a different station of Ana zero because it is a constant minus two is a constant sort himself stayed deal with you. So why this further? Because this is to buy why dash minus two is equal to minus two by dash. So this becomes too by my rash. Plus two y h is equal to two. We can write wired ash y plus two is equal to What about your wired ashes were taken us Common y plus one. This gets cancer, so we have the value wire, dash one. It's different should once again outside. So we have I Double dash will be Will do de or de it, so this can be deterred in those vipers. One race with a bullet minus one so it's different station will be minus, or why plus one restored upon minus true Onda. When we differentiate the stone we get wired ash perceive. So we have minus off one over y plus one who's square. And then we have wide ash in the value of wide ashes. This so people just over here. So we have one of a lifeless one. So the final answer becomes minus one over my plus one. Wait, just about your wired over lunch.


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