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$m$ denotes a fixed nonzero constant, and $c$ is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories.$$y^{2}+m x^{2}=c$$

$y=D x^{1 / m}$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

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in this problem were given. The M is a non zero, but fixed constant. Our goal is to find the orthogonal trajectories. The family of curves, described as y squared plus M X squared equals C, where C is a varying constant. The first step is to take the derivative of all sides of this equation implicitly, due to the fact that the dependent variable has a power of two, so free differentiate implicitly, we obtain that two y times D y DX plus two M times X will be equal to zero. Since the right hand side is constant. Recall that our goal here is to solve for D y t X. So take is our next step. Subtract two MX from both sides. Now two y de y dx equals make to em Times X, and we can divide both sides by two y. So the derivative of why is now equal to negative two mx divide by two y, and upon simplifying, we have that D Y DX is now equal to negative MX divide by why, on the next step, we're going to be able to find those orthogonal trajectories by solving this following differential equation that D Y T X is equal to the negative of the reciprocal of the derivative we just found. So do you. Y DX, in this case, will be equal to Why divide by MX. This differential equation can be solved by first dividing both sides by y. That will give us one over. Why Times D y The X is equal to one over m Times X. Now we could make it a substitution on the left hand side to say that one over. Why Times D. Y. T X is no different than DDX of the natural log of Why Let's give a little bit of space on this line and right that's equal to one over m Times X. Now we conform the anti derivative of both sides of this equation with respect to X and solve. As usual, we're looking for the anti derivative of the derivative of natural log y. So the result is just the natural lager. Their locker them off. Why itself So now the left hand side simplifies to natural log y equals by the constant rule. We have one over m, and now we're looking for the anti drift of of one over X DX, which is known to be the natural log of X plus. The constant of integration will call Upper Casey the next step. Let's use properties of lager of them's allows for one over m to become a exponents on X inside the natural log function. If we do that, the natural log off. Why will be equal to the natural log of X to the power of one over M plus the constant c recalling that our goal is to sulphur. Why, if possible, Russell take as our next step converts the logarithmic equation to a base the exponential equation that gives us y equals e to the power of the natural log of X to the power of one over M plus a constant Let's use next the power rule for exponential to show that Y is equal to eat of the power of C times E to the power of that natural log of X to the power of one over m. The next simplification we make is the fact that the Basie exponential and the natural log a rhythm function are inverse to each other. So we can simplify too. Why equals will call either the sea de and you'll be multiplying extra power of one over M where de is equal to you to the sea. Now this family of curves will be orthogonal to the original given family.

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