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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.$$x^{2}+9 y^{2}=c$$

$\frac{-x}{9 y}$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

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in this problem. We're trying to find the orthogonal trajectories to the families of curves given by the equation X squared plus nine y squared equals C. Our first job in finding those a thaw Grenell trajectories is to differentiate both sides of that equation with respect to X implicitly. So the set up will be that D by D. X of X squared plus nine Weiss Word equals D by the x of the right hand side. See, differentiating a place implicitly on the left gives us two X plus 18. Why? But because Y is a function of X, this will be multiplied by de y dx through the chain rule. The right inside is the driven of a constant and so we obtain just zero. Our next step is to solve for that derivative. D y DX will obtain that 18 y times d y. The X is negative. Two X dividing by 18 y gives us de y dx equals negative two X divide by 18 y or more simply, de y dx equals negative X over nine y, so the slope of the original curves is always equal to negative. X over nine y Barca was to find orthogonal trajectories, and we know from our training and algebra that we find slopes are negative. Reciprocal. Then these curves will be orthogonal. So it's right out that expression to the right that we're seeking a family of curves such that de Y. D X is equal to the reciprocal nine y and the opposite over X for this derivative that we found. Now we have a differential equation that can be organized by first dividing both sides by y we obtain one of her Why de y DX equals nine over X And now we have a helpful trick that tells us that D by D. X of the natural log of y is always equal to one over. Why times de y dx. So it's like use that equation Here we have d by D. X of the natural log of y is now equal to nine over X. This is an equation that could be integrated on both sides, and it would be an integral the left hand side with respect to X as well as integral of the right hand side with respect to X. So it's solved both anti derivatives we know first, the anti derivative of the derivative is just the inter grand itself natural log y. So it's right on the left side. The natural log rhythm of why is now equal to the anti derivative of nine times the natural log of X plus a constant C who were. To solve this differential equation, we can convert to a Basie exponential equation to write that why is now equal to e times or to the power of natural log of X to the power of nine by using properties of logarithms, plus a constant using rules of exponents. This can also be written as why equals e to the power of C Times E to the power of natural log of X to the power of nine. With this arrangement, we also have a cancellation of the natural auger and the function with the Basie Exponential, which allows us to write this solution in the form Why equals? Let's call this constant C Times X to the power of nine, where see is any rial number. So this is the family of curves that's or if agonal to the original family that was given

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