diamond-consider-the-family-of-curves-yk-x2-where-k-is-a-constant-a-show-that-the-differential-equat

Question

$\diamond$ Consider the family of curves $y=k x^{2},$ where $k$ is
a constant.
(a) Show that the differential equation of the family of orthogonal trajectories is
$$
\frac{d y}{d x}=-\frac{x}{2 y}
$$
(b) On the same axes sketch the slope field for the preceding differential equation and several members of the given family of curves. Describe the family of orthogonal trajectories.

Albert Zhang · Accepted Answer

all right. We&#x27;re looking at the family occurs while he goes to see X Square. If we take the differential of both science, we&#x27;re gonna do you Why in close two c x d. Acts like taking the derivative, but then you must you can take out with the axe. Um, so and of course, this is so she Of course. Why don&#x27;t I squared? Which equals a van ness equals two y over acts the axe. So we have X. So you have the slope equal to this every point. So the orthogonal vector field would have a negative inverse over this. That would be the wide equals minus X over two y The axe, which is wife prime, closed, axed over it. You negative acts over. It&#x27;s your wife. This is the differential equation for the year with dog culture factories. Now, if we if we look back here at this equation just equivalent read, we can multiply both sides by two wind to get to wind the wine. You can still see this equation up here to my dear wine equals minus x T ax. So x t x plus to your wife, See why? Because zero So if we integrate this now, we can integrate this when can&#x27;t integrate, why don&#x27;t we get in honor of axity X plus any growth through Why do you? Why? It was a constant. This is one hot X squared Plus why square is he puts or constant? So this is the family or thought Minal curves, um, a tangent lines to this family. Um, so this is a family with our final curse, that better orthogonal to the that they follow the orthogonal trajectory. Tired original Sofield. That&#x27;s where basically what we have is we start out with our family and Paramus. Andi, essentially way get things Family occurs, which is always fun at all. That&#x27;s what I&#x27;m talking about. These problems, it all these points because there were so plans. If this is ellipses, they follow a slow feel, which is orthogonal said that slow field of these proud list. So we get these ellipses that over here and there always one thought at all Is there always right angles? And these ellipses satisfied this kind of equation? Um, they&#x27;re kind of flattened circles. And if we want discussed this little feel, that&#x27;s it just looks like a tangents to these webs is there are other ways to do it like you&#x27;re gonna be control formula, actually, negative X over two y you can see that, for example, on this access, there&#x27;s a on this on the X access we have wired. Crime is equal to zero. So on this access, it&#x27;s all, like best, actually. Here it&#x27;s more like test. Um, because it changes direction here, Um, as it passes through the origin is that there&#x27;s a discounted area, the origin, um which, uh, I can&#x27;t really account for here, but basically the slope you it looks like this construction this low feel like this can sketch these curves. Um that all these parents are talking on. So all these cars, other intersections, you&#x27;re pretty cool. Is there pretty cool? Really Orthogonal curse.

Michael Jacobsen · Answer

in this problem were given a family of curves defined by the equation. X squared plus y squared equals two c x. Our goal is to determine what differential equation do we need to discover in order to find the orthogonal trajectories to that curve. There were to determine that differential equation. We first must differentiate our original equation with respect to X, and that differentiation must be done implicitly, since the dependent variable Y is to a power of two. Let&#x27;s start off. On the left hand side, the derivative of X squared becomes two x. The driven of white scored will give us plus two y times D y the X equals, and now we&#x27;re ready to take the derivative of the right hand side. The independent variable is X has only variable in the right hand side, so he obtained just a to C circle. Next is to solve for D Y DX. We&#x27;ll start off by first subtracting two extra both sides. So now to why Times D Y T X is equal to two C minus two X. Then we could take is our next step to divide both sides of this equation by two y If we do that, we obtained that. D Y T X is equal to si divide by why minus X divide by Why, since the factor of two cancels throughout as our next step, we need to note that the constant C depends on both Excel y at the same time due to the original equation. So it&#x27;s taken extra step to rewrite this equation. X squared plus y squared equals two c x and then solve it for the variable. See, we divide both sides by two x we obtained that see is now equal to X squared plus y squared divide by two X, and that substitution needs to be made here. Let&#x27;s go back to this original derivative for why it cannot be expressed as D y DX equals one over why times The expression for C and C is X squared, plus y sward divide by two x so that finishes the substitution. And let&#x27;s just copy a negative X over why The next thing we ought to work with on this equation is to find its most simplest form. With between the two fractions, we can simplify by including a factor of two in the numerator denominator as well as a factor of X. So put in the denominator, too, as well as X numerator. We have to and multiply by X. Then this would simplify too. X squared plus y squared divide by two x y from the first fraction minus two X squared Divide by two ex wife from the second fraction. If we subtract across the numerator, we&#x27;ll obtain that the numerator is y squared minus X squared. Divide by two x y every prefer X squared. Coming first weaken. Factor out a negative sign from the numerator to obtain negative of X squared minus y squared. Divide by two X y is the slope to the family of equations. So now we&#x27;re ready to state our solution. The answer to this problem is that the differential equation is found by writing de y over DX and taking the negative reciprocal off the last expression we just found. So our differential equation will read d Y d X equals positive two X y divide by X squared minus y squared. So that solves this problem

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answer to question number 12 from chapter 16.3. So a for us to find the family of solutions off this different to the question on let&#x27;s find the family of all doctoral trajectories. First, to find the solutions, we can just, ah separate the variables into X. The X is equal to minus why you why and integrate both sides. We need to add a constant of integration. We&#x27;re gonna have half X squared. Let&#x27;s put the wise and on the left inside as well. Plus 1/2 why? Swears is able to see let me just most by both sides by two and absorb the factor two into C because it&#x27;s just are pretty constant. Who got X where y squared plus X squared equal to see. So we&#x27;re now we need to find the authority of trajectories at this. Thank you. Why the X for these curves? He&#x27;s gonna be acquitted minus X over. Why it took me to see that straight from the difference equation that were given so that for the All Star you know, trajectories, we&#x27;re gonna have why the X equal to the negative Pacifica with this, which is why of X. So we could separate the very was again. You got one of why why equals in school of one of X dx, plus a constant of integration? Do you have the natural log? Why is equal to the natural log of X? We could take E to the power of both sides. So it&#x27;s essential that both sides, they even have why is equal to a constant a Times X sees your straight lines through the origin so you can see that the solutions we&#x27;re circles because they were X squared plus y squared is equal toe. See, So the seconds would various radio radio I on the orthogonal trajectories, straight lines through the origin like second, I&#x27;m not be whereas to find these lesions of this defense situation a separate the very bulls and we&#x27;re gonna have whatever. Why why is equal to come on to time is one of X, the X It&#x27;s not integrate both sides, not a constant of integration. That natural log of why is equal to two side the natural log of X. Let&#x27;s see take exponential is both sides we get why is equal, Teoh constant a time is next. So the grading of these solutions you by the X is equal Teoh a X So therefore orthogonal trajectories. We&#x27;ll have Grady in Do you want? The X is equal to the negative reciprocal of this so minus two at minus one over to a X. Then integrating this, we&#x27;re gonna have why is equal to minus one over to A. It&#x27;s just a constant size. The natural log, the absolute value of X plus another constant integration. Let&#x27;s call it D. So now this of has for a given a gives the authority trajectories to the south curves getting here.

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

Michael Jacobsen · Answer

in this problem. We&#x27;re looking for the thaw gurnal trajectories to the curve given to be why equals C times the exponential function each of the X. The first step is to differentiate both sides of that equation. So we&#x27;ll have that D y t x will be equal to the derivative of a constant C times an exponential What that would give us C times either the ex itself. Since our exponential is to the base e To find the full expression for this derivative, we need to substitute out the constant C since see, it depends on both X and Y from the original family of curves. We know that C is equal to why divide by either the ex. So we&#x27;ll substitute that back into our derivative for our next step. D y d X is now equal to in place of see we have why divide by you to the X times, even the X. That gives us a quick cancellation in this problem, or the base of the X is cancelled from the numerator denominator. And now we have that D y d X is equal to why itself next to find orthogonal trajectories, We&#x27;ll take that D Y DX must be equal to the negative reciprocal of the expression we have just found. So our differential equation becomes D Y D X equals negative one over why and to solve this differential equation, let&#x27;s collect all portions that involved why toe left inside and all the portions involved x to the right hand side. Upon doing that, we have that why d y will be equal to negative one times DX. Then we can integrate both sides the left hand side with the respect to why the right hand side with respect to X to obtain that 1/2 a y squared is equal to negative X, plus a constant. Let&#x27;s call this one upper case C. If we add extra pill sides, the equation is now X plus. 1/2 of y squared is equal to upper case C, and then we can simplify this equation further by multiplying both sides by two. If we do that, the equation becomes two. X plus y squared equals a constant will call this one D where d equals two times Upper Casey, and this is the equation for the orthogonal trajectories off the original family

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

O Consider the differential equation $$ \frac{d i…

Problem 38 Hard Difficulty

Answer

$\frac{d y}{d x}=-\frac{x}{2 y}$

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