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Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family.$$y=c x^{2}$$

$\frac{ }{}$ $x^{2}+2 y^{2}=D$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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in this problem, we're looking for their orthogonal trajectories to the equation. Y equals C times X squared. We start by differentiating both sides of this equation implicitly. With respect to X, the left hand side will be de y DX, and the right hand side will be too time. See Time's X to the power of one fix expression for why Prime or D y DX involves the constant C. But see himself depends on both the function. Why, as well as an independent variable X buff we saw for C, We know that C is equal to why divide by X squared. Let's substitute that back into our expression for the derivative D y. The X we'll obtain that d y. The X is equal to two times of expression for C, which is why divide by X squared times X and upon simplifying this right hand side of the derivative D Y DX, well obtained to why divide by X, the next step of this process is to now solve the differential equation de y dx, which is equal to taking their original expression for the derivative and multiplying it by negative one. As well as taking the reciprocal. So we have negative X divide by two y is equal to D y DX. Now the salt this differential equation. I'm going to collect all expressions involved. Why the D. Y and the two y and all expressions involved X and bring them to opposite sides of this equation will obtain to why de y is equal to negative x DX. And if we integrate both sides of this expression, then we'll obtain that Weiss word is equal to negative 1/2 of X squared, plus a constant of integration. Let's call this one upper case C. The equation is essentially solved, but we can simplify it further by adding 1/2 X squared double sides. So half X squared plus y squared equals a constant C. If we'd like to simplify further, multiplying both sides of the equation by two gives us the family of curves. Ex word plus two y squared equals. Let's use a new constant D where de is equal to two time see, and this is the format for the family of curves that's orthogonal to the original family. Given

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