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$\diamond$ Consider the family of curves $y=k x^{2},$ where $k$ isa 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.

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

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 3

The Geometry of First-Order Differential Equations

Differential Equations

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

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04:31

Consider the family of cir…

05:05

Find the family of solutio…

02:57

Find the equation of the o…

02:53

08:24

Determine the orthogonal t…

09:36

03:04

03:35

Find the orthogonal trajec…

01:45

06:08

all right. We're looking at the family occurs while he goes to see X Square. If we take the differential of both science, we'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'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'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't integrate, why don'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'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's what I'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're kind of flattened circles. And if we want discussed this little feel, that's it just looks like a tangents to these webs is there are other ways to do it like you're gonna be control formula, actually, negative X over two y you can see that, for example, on this access, there's a on this on the X access we have wired. Crime is equal to zero. So on this access, it's all, like best, actually. Here it's more like test. Um, because it changes direction here, Um, as it passes through the origin is that there's a discounted area, the origin, um which, uh, I can'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're pretty cool. Is there pretty cool? Really Orthogonal curse.

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