find-the-orthogonal-trajectories-of-the-family-of-curves-sketch-several-members-of-each-family-yc-x2

Question

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$y=c x^{2}$$

Matthew Allcock · Accepted Answer

in this video, we&#x27;re gonna go to the Ansett&#x27;s question about six, which have a 16.3 to ratify the orthogonal trajectories to the family of curves and my wise you want to see X squared? Let&#x27;s find the brilliant off this from your curves. So just differentiating there&#x27;s gonna be to see of X so that for uh oh, I thought, you know, trajectories are gonna have Grady. And given by the Y, the X is equal to negative is typical of this so we can integrate that. Ah, see, we can interpret that using separation variables Philippines great left inside like so I would take a a constant minus 1/2. I wanted to see as integral one of acts the X And let&#x27;s add to that a constant of integration was Call it G integration Left inside, we get why that&#x27;s equal to minus one over to see integrate one over X and you get the natural log of X. That&#x27;s like the great Innovex. Sorry, the National Ogilvy Absolute value of X plus day as that&#x27;s the, uh, all the books directories

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answer to a question. About eight jumped 16.3 arrested. Find the family orthogonal trajectories to the curve. What the family of curves, given by two x squared plus y squared equals C squared. So it&#x27;s most. Differentiate this with respect to X to find the greedy INTs of this curve. Said to find d y DX difference in the first term, Get four x The second term we get to Why do you want the X is in the general at zero on the right side? Because it&#x27;s constant. It&#x27;s that funny why the X could be able to minus two x over. Why so that for the orthogonal trajectories have radiance d y the X. Which of negative is typical of this? Which is why over two X we can then so this inseparable different equation by integrating one over Why? Let&#x27;s back to why. Let us be equal, Teoh, I say take out Calmund. So I take out the constant of 1/2 outside of the integral get one of eggs. The X plus. It&#x27;s called the constant D, so that gives us not to log. Why is equal to ha times the natural log backs plus day. So taking either power of both sides looking left inside to get why is equal to then we&#x27;re gonna have I was gonna be eat the d which we can just call a new constant times by seeds And Dr Long Next Carvajal I was just gonna be extend Borrow ha! Who will rewrite this so that it looks more like what we expect a hyperba Let&#x27;s look like is gonna be X is equal to Let&#x27;s call the a one a new constant widespread.

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answer. It&#x27;s question of benign from chapter 16.3. So for us to find the orthogonal trajectories, the family of curves given by Y is equal to see a constant times e minus x. So let&#x27;s differentiate this. Expect X to find out what the grading of this kind of is. Dwight D. X. You have equal Teoh minus c times e sex. The orthogonal projector is I&#x27;m gonna have girls getting by the negative, reciprocal reckless to do y the x. It was equal to one of the sea times e to the X. So just integrate this to find why is equal to one of the sea He&#x27;s the ex Aziz the actress integrates too. So plus a constant let&#x27;s call it be

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answer to question number 10 jumped 16.3 to rest if I into the orthogonal trajectories of the family of curves. Why is equal to eat the K X work? A is a constant diverse. Once find the Grady in off the cars is equal to u k E X. We know that the author no trajectories are gonna have ingredient Pribadi X given by the next reciprocal this That&#x27;s equal. Teoh minus one of Okay, tons e tu minus kayaks. We can solve this just by direct integration. You&#x27;re gonna have Why is equal to minus one of the K that we&#x27;re gonna have to times that by minus one of the K E to the minus k X plus a constant. Let&#x27;s call it. Let&#x27;s go. See, that&#x27;s one of the case squared E to the minus K X. Let&#x27;s see

Michael Jacobsen · Answer

in this problem, we&#x27;re looking for their orthogonal trajectories to the equation. Y equals C times X squared. We start by differentiating both sides of this equation implicitly. With respect to X, the left hand side will be de y DX, and the right hand side will be too time. See Time&#x27;s X to the power of one fix expression for why Prime or D y DX involves the constant C. But see himself depends on both the function. Why, as well as an independent variable X buff we saw for C, We know that C is equal to why divide by X squared. Let&#x27;s substitute that back into our expression for the derivative D y. The X we&#x27;ll obtain that d y. The X is equal to two times of expression for C, which is why divide by X squared times X and upon simplifying this right hand side of the derivative D Y DX, well obtained to why divide by X, the next step of this process is to now solve the differential equation de y dx, which is equal to taking their original expression for the derivative and multiplying it by negative one. As well as taking the reciprocal. So we have negative X divide by two y is equal to D y DX. Now the salt this differential equation. I&#x27;m going to collect all expressions involved. Why the D. Y and the two y and all expressions involved X and bring them to opposite sides of this equation will obtain to why de y is equal to negative x DX. And if we integrate both sides of this expression, then we&#x27;ll obtain that Weiss word is equal to negative 1/2 of X squared, plus a constant of integration. Let&#x27;s call this one upper case C. The equation is essentially solved, but we can simplify it further by adding 1/2 X squared double sides. So half X squared plus y squared equals a constant C. If we&#x27;d like to simplify further, multiplying both sides of the equation by two gives us the family of curves. Ex word plus two y squared equals. Let&#x27;s use a new constant D where de is equal to two time see, and this is the format for the family of curves that&#x27;s orthogonal to the original family. Given

Banhishikha Sinha · Answer

Well, here we are solving the push Number 30 We have given X squared life. Boy square? Yes, they called stool. I don&#x27;t see a way that&#x27;s differently. They questions implicitly with respect to X, which will be two x less so. Why the life? My DX is because you to see. Do you like baby? It&#x27;s becomes the wife. Yes, X My guess why market as a question one? This contains constant C So it is not the difference immigration off the given family so we can express the value off X on the bill in question. But this this one so you can read it is see is a question exclude? No. Why? Square divided by the way. Now substituting this an equation one you get Oh, well, why my two years? Yes, he called Stool x. A group? Yes. Why Square do Why? I mean, it&#x27;s a tropical store X squared plus y squared to square Modify boy, which becomes door. That&#x27;s why stroll square minus my square. From this equation, we get the difference Still equation for the family of portable tragic story. This is the wife ideas The coastal my Nestle exclude minus y script divided by dual X y. Can the witness do you? Why my idea? There&#x27;s pickles toe minus one minus. Why buy Hex? We go. Do you know why? Thanks now Substrate puting. Why speak with stool? Thanks. Be thanks. You convey d y by D it? Yes. Equals to be less x b by the X we&#x27;ve close. Do you be by D X close to one, minus this square by Do we? No, thanks. Nick will be on the right side. Next. Davies idea. Nicholas, stool, maybe square my next one after it so operated form you will be by three b squared minus one. This sheik was still give me a yes. It goes to the X by X now in to get both the sites of the question left side by B and they would say by X like the some instigation will be three square in this one. They request to integration one by eggs in your smart get is in question Took so the integral off this side it is the left side. Let&#x27;s put previous governments one is equals to think which we come six me Do you me some cold stew about into girls becomes intimidation. My B Dave Days Stool The Natural Law Day. Let&#x27;s see. Stop shooting back. The value for these questions do becomes three. Natural law No. Three Squared minus one. Is it still natural? Next messy that comes through this gloominess. One. We&#x27;ll kill. We&#x27;ll still see one. Knicks No substituting back or B is a cold too. Live by s. We get dream. Why buy eggs? Hold square minus one. My cue Because do anyone necks? No, this is cool. 23 way square. My news It&#x27;s good. Hope you close to see line to the bar for So this is a question of the family of portable and tragic stories. Oh, next let&#x27;s drew it. This is 100 another 100 No y Axis 200. This 100 someone to use Bring one for the family of what I wanna trajectories for the oneness because to 90 which way? Who and these and I&#x27;m raising that one for six was 2 20 True. Yeah. Move this. No

Problem

Find the orthogonal trajectories of the family of…

Problem 6 Easy Difficulty

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
$$y=c x^{2}$$

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Joel Hass, Christopher Heil, Przemyslaw BogackiUniversity Calculus: Early Transcendentals

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