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Question

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

Matthew Allcock · Accepted 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

Matthew Allcock · 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

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

Matthew Allcock · Answer

in this video, we&#x27;re gonna go through the answer to question number five from chapter 16.3. So testifying to the orthogonal trajectories off the curve, why is it was m facts where m is a constant. So curves Why was m X are gonna have Grady in the Y The x of i m so that four. Well, I thought, you know, trajectories, I&#x27;m gonna have Grady in. You are The X is equal to negative Pacific, all minus one of them. Integrate this to find the family of curves that&#x27;s gonna give us why is equal to minus one of them ex plus a constancy.

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.

Problem

Find the orthogonal trajectories of the family of…

Problem 9 Easy Difficulty

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

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