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Find the orthogonal trajectories of the family of curves. Sketch several members of each family.$$k x^{2}+y^{2}=1$$

Calculus 2 / BC

Chapter 16

First-Order Differential Equations

Section 3

Applications

Differential Equations

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University of Michigan - Ann Arbor

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in this video, we're gonna go through the answer to question number seven from jumped 16.3 to rest to find the orthogonal trajectories to the family off curves Given my k X squared plus y squared is equal to one. Did everyone see differentiate this equation with speculative divine d y dx different photo differentiation the first time we get to X? Okay, the second time we get to Why do you want the X is using the train will for that term Example 20 Okay, so these curves have radiant given by do why the X is equal to minus K x over why lots of that Foley on the orthogonal trajectories I have great and given by do why the X is equal to the negative is typical of this. That's gonna be why of a K X. This is a separable differential equation. So we can so bit by integrating one over. Why do you Why equal Teoh one of a k and take our constant in school of one of X. The X was the constancy. It's great one over why you got the natural log we actually value of why and again integrate one of expect the absolute value. It does not look of the absolute value of X plus C so we can take Eastern power both sides it's the left hand side is easy. That's just gonna be why the right hand side we can write this is e to the sea times e to the natural log Thanks. The power of one of the K so eats the scene. We can just call up that you know, the name is called Capsule A for A and you constant than e and natural log invest functions cause this is just gonna bay X to the power of one of the K So that describes our family of curves. We could write that in a different way, as X is equal to a let's call a one quality from constant times Why, okay,

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