Problem

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

03:18

Problem 18 Easy 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.
$$y=m x+c$$

Answer

$\frac{x}{m}+y=c$

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

for this problem were given that M is a fixed but non zero constant. A miracle is then to find the orthogonal trajectories to the curve, y equals MX plus e. To do that, we start off by taking the derivative of Why so D y DX would become M itself then the next stage of this process is now to solve the differential equation that D Y DX is the negative, reciprocal off the derivative we've just found, So D Y T X is negative one over m. What we need to do next is integrate both sides of this equation with respect to X. Let's start off by re copying the equation with a little bit of space. So D Y T X is equal to negative one over m. Then integrate the left hand side with respect to X as well as the right hand side. Also with respect to X. If we look at the left hand side, it's asking for the anti derivative of the drift of of why? What survives is just why itself, which will be equal to negative one over m times the variable X plus a constant C next. If we add negative one over M times extra bull sides of this equation. We obtain X divide by m plus. Why is equal to see and this represents the family of orthogonal trajectories to the original family of curves.

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