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Find the family of solutions of the given differential equation and the family of orthogonal trajectories. Sketch both families.a. $x d x+y d y=0$b. $x d y-2 y d x=0$

Calculus 2 / BC

Chapter 16

First-Order Differential Equations

Section 3

Applications

Differential Equations

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in this video, we'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'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're gonna have half X squared. Let'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's just are pretty constant. Who got X where y squared plus X squared equal to see. So we're now we need to find the authority of trajectories at this. Thank you. Why the X for these curves? He'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'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'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'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'm not be whereas to find these lesions of this defense situation a separate the very bulls and we're gonna have whatever. Why why is equal to come on to time is one of X, the X It'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'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'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're gonna have why is equal to minus one over to A. It's just a constant size. The natural log, the absolute value of X plus another constant integration. Let's call it D. So now this of has for a given a gives the authority trajectories to the south curves getting here.

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