Problem

$m$ denotes a fixed nonzero constant, and $c$ is …

01:39

Problem 17 Medium Difficulty

$m$ 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.
$$y=c x^{m}$$

Answer

$x^{2}+m y^{2}=D$

Related Courses

Calculus 2 / BC

Differential Equations and Linear Algebra

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

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Video Transcript

in this problem, we're giving a fixed but non zero constant m. Our goal here is to find the orthogonal trajectories to this family of curb curbs. Wyche will see times X to the power of them to start off will take the derivative of why and obtain that d y dx by the power rule first. Well, right that this is equal to M times C times X to the power of M minus one through the power rule. Our next step in this process is to determine the value for that constant C since see depends on both y and X at the same time. If we divide by X to the power of them, we see that C is equal to why divide by X to the power of them and we can substitute this expression back into our derivative. So for the next step, de y, the X by substitution is equal to em times. Why divide by X to the power of them Times X to the power of and my s one. Let's simplify this further to show that d y the X is now equal to m times. Why divide by X Our next step now is to find the family of orthogonal trajectories that could be solved by writing the differential equation. De Y. D X is equal to the negative reciprocal of the derivative that we have just found, which had become a negative X divide by and why we consult this differential equation by bringing white together and white together with D Y and X, together with DX, the differential equation that becomes m y times d y equals negative x times, DX. Then we can integrate the left hand side with respect to why and the right hand side with respect X. This produces M Divide by two y squared equals a negative X squared divide by two, plus a constant of integration. Let's go with uppercase. See for that constant. If we like, we can add X squared divide by two to both sides of this equation. To give the slightly nicer format X squared over two plus m over two y squared equals Upper Casey. Then, if we multiply the entire equation by two, we get the result that X squared plus M y squared is equal to D, where he is equal to twice C. Then this is our family or auth organ, all trajectories to the original curve that was given

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